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NBER WORKING PAPER SERIES
INTERNATIONAL FRIENDS AND ENEMIES
Benny KleinmanErnest Liu
Stephen J. Redding
Working Paper 27587http://www.nber.org/papers/w27587
NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue
Cambridge, MA 02138July 2020
We are grateful to Princeton University for research support. We would like to thank Gordon Ji and Ian Sapollnik for excellent research assistance. We are also grateful to Costas Arkolakis, Dave Donaldson, Jonathan Eaton, Andrés Rodríguez-Clare, Shang-Jin Wei, Daniel Xu and conference and seminar participants at Nottingham, National Bureau of Economic Research (NBER), and Princeton for helpful comments and suggestions. We are grateful to Robert Feenstra and Mingzhi Xu for generously sharing updates of the NBER World Trade Database. The usual disclaimer applies. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.
© 2020 by Benny Kleinman, Ernest Liu, and Stephen J. Redding. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
International Friends and EnemiesBenny Kleinman, Ernest Liu, and Stephen J. ReddingNBER Working Paper No. 27587July 2020JEL No. F14,F15,F50
ABSTRACT
We develop sufficient statistics of countries' bilateral income and welfare exposure to foreign productivity shocks that are exact for small shocks in the class of models with a constant trade elasticity. For large shocks, we characterize the quality of the approximation, and show it to be almost exact. We compute these sufficient statistics for over 140 countries from 1970-2012. We show that our exposure measures depend on market-size, cross-substitution and cost of living effects. As countries become greater economic friends in terms of welfare exposure, they become greater political friends in terms of United Nations voting and strategic rivalries.
Benny KleinmanDepartment of EconomicsJulis Romo Rabinowitz BuildingPrinceton, NJ [email protected]
Ernest LiuPrinceton UniversityDepartment of EconomicsJulis Romo Rabinowitz BuildingPrinceton, NJ [email protected]
Stephen J. ReddingDepartment of Economics &School of Public and International AffairsPrinceton UniversityPrinceton, NJ 08544and CEPRand also [email protected]
1 Introduction
One of the most dramatic changes in the world economy over the past half century has been the emergence of China
as a major force in world trade. A central question in international economics is the implications of such economic
growth for the income and welfare of trade partners. A related question in political economy is the extent to which
these large-scale changes in relative economic size necessarily involve heightened political tension and realignments
in the international balance of power. We provide new theory and evidence on both of these questions by developing
bilateral “friends” and “enemies” measures of countries’ income and welfare exposure to foreign productivity shocks
that can be computed using only observed trade data. Our measures are derived directly from the leading class of
international trade models with a constant trade elasticity, are computationally trivial to compute, and have a clear
economic interpretation even in rich quantitative settings with many countries. We show that our analysis admits
a large number of generalizations, including multiple sectors, input-output linkages and factor mobility (economic
geography). We derive bounds for the sensitivity of countries’ exposure to foreign productivity shocks to departures
from a constant trade elasticity. Using standard matrix inversion techniques, we compute over 1 million bilateral
comparative statics for income and welfare exposure to foreign productivity growth for more than 140 countries from
1970-2012 in a few seconds. Consistent with the idea that con�icting economic interests can spawn political discord,
we �nd that as countries become greater economic friends in terms of the welfare e�ects of their productivity growth,
they also become greater political friends in terms of the similarity of their foreign policy stances, as measured by
United Nations voting patterns and strategic rivalries.
Our research contributes to the recent revolution in international trade of the development of quantitative trade
models following Eaton and Kortum (2002) and Arkolakis, Costinot and Rodriguez-Clare (2012). A key advantage of
these quantitative models is that they are rich enough to capture �rst-order features of the data, such as a gravity
equation for bilateral trade, and yet remain su�ciently tractable as to be amenable to counterfactual analysis, with a
small number of structural parameters. A key challenge is that these models are highly non-linear, which can make
it di�cult to understand the economic explanations for quantitative �ndings for particular countries or industries.
A key contribution of our bilateral friends-and-enemies measures is to reveal the role played by di�erent economic
mechanisms in these models. In particular, we show that the e�ect of a productivity shock in a given country on
welfare in each country depends on three matrices of observed trade shares: (i) an expenditure share matrix (S) that
re�ects the expenditure share of each importer on each exporter; (ii) an income share matrix (T) that captures the
share of each exporter’s value added derived from each importer; (ii) a cross-substitution matrix (M) that summarizes
how an increase in the competitiveness of one country leads consumers to substitute away from all other countries
in each market. Using these results, we separate out countries’ welfare exposure to foreign productivity shocks into
income and cost-of-living e�ects; break out income exposure to these productivity shocks into market-size and cross-
substitution e�ects; isolate partial and general equilibrium e�ects; evaluate the contributions of individual sectors to
aggregates; and assess the contribution of importer, exporter and third-market e�ects.
Our bilateral friends-and-enemies measures are exact for small productivity shocks for this leading class of in-
ternational trade models characterized by a constant trade elasticity. For large productivity shocks, we provide two
sets of analytical results for the quality of our approximation. First, we compare our linearization to the non-linear
exact-hat algebra approach that is commonly used for counterfactuals in constant elasticity trade models. We show
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that the quality of our approximation depends on the properties of the observed trade matrices (S, T and M). Given
the observed values of these matrices and productivity shocks of the magnitude implied by the observed trade data,
we �nd that the two sets of predictions are almost visibly indistinguishable from one another, with a regression slope
and R-squared close to one. Second, we compare our results for a constant trade elasticity with those for a variable
trade elasticity, and derive sensitivity bounds for the impact of productivity shocks in this more general speci�cation
with a variable trade elasticity. As such, our characterization of the incidence of productivity and trade shocks in
terms of the market-size, cross-substitution and cost-of-living e�ects provides a useful benchmark for interpreting
the results of quantitative trade models outside of our class.
Our main empirical contribution is to use our friends-and-enemies exposure measures to examine the global in-
cidence of productivity growth in each country on income and welfare in more than 140 countries over more than
forty years from 1970-2012. We �nd a substantial and statistically signi�cant increase in both the mean and dispersion
of welfare exposure to foreign productivity shocks over our sample period, consistent with increasing globalization
enhancing countries’ economic dependence on one another. We �nd that productivity growth in most countries raises
their own income compared to world GDP and reduces the income of most (but not all) other countries compared to
world GDP. Even compared to a weighted average of OECD countries, we �nd that Chinese productivity growth has
an increasingly large negative e�ect on US relative income. Nevertheless, once changes in the cost of living are taken
into account, this Chinese productivity growth has an increasingly large positive e�ect on aggregate US welfare. More
generally, we provide evidence of large-scale changes in bilateral patterns of welfare exposure to foreign productivity
growth following trade liberalization in North America, the fall of the Iron Curtain in Europe, and the replacement of
Japan by China at the center of geographic production networks in Asia.
We decompose overall income and welfare exposure into the direct (partial equilibrium) e�ect of foreign produc-
tivity at initial incomes and the indirect (general equilibrium) e�ect through endogenous changes in incomes. We �nd
that these general equilibrium forces are quantitatively large in this class of models, such that misleading conclusions
about the income and welfare e�ects of productivity growth can be drawn from simply looking at the partial equi-
librium terms alone. We �nd that both the cross-substitution e�ect and the market-size e�ect make substantial con-
tributions to overall income exposure. On the one hand, the partial equilibrium component of the cross-substitution
e�ect is always negative, because higher productivity growth in a given country increases its price competitiveness
and leads to substitution away from all other countries. On the other hand, the general equilibrium components of the
cross-substitution and market-size e�ects can be either positive or negative, because higher productivity in a given
country raises its own income and a�ects income in all other countries, which in turn induces changes in price com-
petitiveness and market demand for all countries. Finally, although much of the e�ect of foreign productivity growth
on home income occurs through the home country’s market, we also �nd a substantial e�ect through the foreign
country’s market, and empirically-relevant e�ects through the home country’s most important third markets.
We compare our friends-and-enemies exposure measures in our baseline model with a single sector to those in
models with multiple sectors and input-output linkages. Although there is a strong correlation between the predic-
tions of all three models, we �nd that introducing both sectoral comparative advantage and production networks
has quantitatively relevant e�ects on bilateral income and welfare exposure for individual pairs of exporters and im-
porters. Additionally, both the multiple-sector and input-output models yield additional disaggregated sector-level
predictions, in which even foreign productivity growth that is common across sectors can have heterogeneous e�ects
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across individual industries in trade partners, depending on the extent to which countries compete with one another
in sectoral output markets versus source intermediate inputs from one another. Comparing these sector-level predic-
tions for the impact of Chinese productivity growth, we �nd some marked di�erences across countries. For nearby
South-East Asian countries, the sectors that bene�t most include the Electrical, Medical and O�ce sectors, consistent
with input-output linkages between related sectors through global value chains in Factory Asia. However, for the
resource-rich emerging economies, the sectors that bene�t most include the Mining, Agricultural and Basic Metals
sectors, consistent with a form of “Dutch Disease,” in which the growth of resource-intensive sectors propelled by
Chinese demand competes away factors of production from less resource-intensive sectors.
We use our friends-and-enemies exposure measures to provide new evidence on a political economy debate about
the extent to which increased economic rivalry between nations necessarily involves heightened political tension. A
number of scholars have drawn parallels between the current China-US tensions and earlier historical episodes, such
as the confrontation between Germany and Great Britain around the turn of the twentieth century, and the rise of
Athens that instilled fear in Sparta that itself made war more likely (the Thucydides Trap).1 On the one hand, there
are good reasons to be skeptical about this essentially mercantilist view of the world, because a key insight from trade
theory is that trade between countries is not zero-sum. On the other hand, it remains possible that the extent to which
countries have shared economic interests is predictive of their political alignment. Consistent with this view, we �nd
that as countries become less economically friendly in terms of the welfare e�ects of their productivity growth, they
also become less politically friendly in terms of their foreign policy stances, as measured by United Nations voting
patterns and strategic rivalries.
Our research is related to several strands of existing work. First, traditional neoclassical theories of trade highlight
the terms of trade as the central economic mechanism through which shocks to productivity, transport costs and trade
policies a�ect welfare in other countries. Key insights from this literature are that foreign productivity growth can
either raise or reduce domestic welfare depending on whether it is export or import-biased (e.g. Hicks 1953, Johnson
1955 and Krugman 1994), and immiserizing growth becomes a theoretical possibility if domestic productivity growth
induces a su�ciently large deterioration in the terms of trade (see Bhagwati 1958). While this theoretical literature
isolates key economic mechanisms, the empirical magnitude of these e�ects remains unclear, because these theoretical
results are typically derived in stylized settings with homogeneous goods and a small number of countries and goods
(typically two countries and two goods).
Second, we contribute to the growing literature on quantitative trade models following the seminal and Frisch-
medal winning research of Eaton and Kortum (2002), including Dekle et al. (2007), Costinot et al. (2012), Caliendo
and Parro (2015), Adão et al. (2017), Burstein and Vogel (2017), Caliendo et al. (2018), and Levchenko and Zhang
(2016). Using a multi-sector quantitative trade model, Hsieh and Ossa (2016) �nd small spillover e�ects of Chinese
productivity growth from 1995-2007 on other countries’ welfare, which range from -0.2 percent to 0.2 percent. In
a counterfactual analysis of alternative patterns of Chinese productivity growth, di Giovanni et al. (2014) �nd that
most countries experience larger welfare gains when China’s productivity growth is biased towards comparative
disadvantage sectors. In a speci�cation incorporating many local labor markets within the United States, Caliendo
et al. (2019) develop a quantitative trade model that replicates reduced-form empirical �ndings for the China shock,1See for example Brunnermeier et al. (2018) and “China-US rivalry and threats to globalisation recall ominous past, ” Martin Wolf, Financial
Times, 26th May, 2020.
3
with net welfare gains for the United States as a whole, but heterogeneity across local labor markets. We contribute
to this research by developing new friends and enemies measures of exposure to foreign productivity growth that
closely replicate the full nonlinear solution of the leading class of quantitative trade models, while also revealing the
role of the key economic mechanisms through which productivity growth in one country a�ects welfare in another
in these models. The low computational burden of our approach lends itself to applications in which large numbers
of counterfactuals must be undertaken, as in our empirical application with more than 1 million comparative statics.
Furthermore, the wide range of extensions and generalizations of our approach allow researchers to easily compare
and contrast the results of large numbers of counterfactuals across di�erent quantitative models, such as our single-
sector, multi-sector and input-output speci�cations.
Third, our work is related to the burgeoning literature on su�cient statistics for welfare in international trade,
including Arkolakis et al. (2012), Caliendo et al. (2017), Baqaee and Farhi (2019), Galle et al. (2018), Huo et al. (2019),
Bartelme et al. (2019), and Kim and Vogel (2020). Within a class of leading international trade models, Arkolakis et al.
(2012) shows that the welfare gains from trade can be measured using only a country’s domestic trade share and a
constant trade elasticity. In a wider class of gravity models, Allen et al. (2020) use the network structure of trade to
prove existence and uniqueness, and show that counterfactual predictions in this class of models have a series expan-
sion representation in terms of demand and supply matrices that are functions of trade data and demand and supply
elasticities. In a model with general spatial links between local labor markets, Adão et al. (2019) characterize general
equilibrium elasticities of employment, wages, and real wages in each market with respect to shift-share measures of
exposure to foreign trade shocks using revenue and consumption shares. In a general network economy, Baqaee and
Farhi (2019) derive �rst-order counterfactual formulas (and second-order accounting formulas) for productivity and
trade shocks, and implement these for a nested CES economy. Our main contribution relative to this body of research
on su�cient statistics is to derive bilateral friends-and-enemies measures of exposure to foreign productivity shocks
that can be directly connected to underlying market-size, cross-substitution and cost of living e�ects in a large class of
quantitative trade models. We show that these friends-and-enemies measures can be recovered from observed income
and expenditure share data using standard matrix inversion techniques. Although our measures capture �rst-order
general equilibrium counterfactuals, we provide an analytical characterization of the magnitude of the second and
higher-order terms, and hence the quality of our approximation to the full non-linear model solution.
Fourth, our research connects with the large reduced-form literature that has examined the domestic e�ects of
trade shocks (such as the China shock), including Topalova (2010), Kovak (2013), Dix-Carneiro and Kovak (2015),
Autor et al. (2013), Autor et al. (2014), Amiti et al. (2017), Pierce and Schott (2016), Feenstra et al. (2019), Borusyak
and Jaravel (2019), and Sager and Jaravel (2019). A key contribution of this empirical research has been to provide
compelling causal evidence on the e�ects of trade shocks using quasi-experimental variation. A continuing source of
debate in implementing this empirical analysis is the appropriate measurement of trade shocks, including whether to
focus on imports from one country, a group of countries or all countries; how to capture imports of �nal goods versus
intermediate inputs; how to incorporate exports as well as imports; and how to measure third-market e�ects. Our
research contributes to this debate by deriving theory-consistent measures of productivity and trade costs shocks that
use only observed trade data, and that capture all of the above channels, including both partial and general equilibrium
e�ects. As these su�cient statistic measures are derived from a class of theoretical models, they yield predictions for
model-based objects such as welfare as well as for observed variables such as income.
4
Fifth, our analysis of countries’ bilateral political attitudes is related to a large literature in economics, history and
political science, including Scott (1955), Cohen (1960), Signorino and Ritter (1999), Alesina and Spolaore (2003), Martin
et al. (2008), Kuziemko and Werker (2006), Guiso et al. (2009), Bao et al. (2019), and Häge (2011). We use our measures
of exposure to productivity shocks to provide new evidence on the classic political economy question of the extent to
which countries with shared economic interests also have similar political stances.
The remainder of the paper is structured as follows. Section 2 provides a characterization of the e�ects of produc-
tivity shocks in each country on income and welfare in all countries in an Armington model with a general homothetic
utility function. Section 3 develops our measures of countries’ income and welfare exposure to foreign productivity
shocks for the special case of this model that falls within the class of models with a constant trade elasticity considered
by Arkolakis et al. (2012), henceforth ACR. Section 4 reports a number of extensions and generalizations, including
trade imbalances, small departures from a constant trade elasticity, multiple sectors following Costinot et al. (2012),
henceforth CDK, and input-output linkages following Caliendo and Parro (2015), henceforth CP, and economic geog-
raphy models with factor mobility. Section 5 reports our main empirical results for the impact of a productivity shock
in one country on income and welfare in all countries. Section 6 provides empirical evidence on the extent to which
countries that are economic friends of one another are also political friends. Section 7 concludes. A separate online
appendix contains the derivations of the results in each section of the paper and the proofs of the propositions.
2 General Armington
We consider an Armington model with a general homothetic utility function, in which goods are di�erentiated by
country of origin. We consider a world of many countries indexed by n, i ∈ {1, . . . , N}. Each country has an
exogenous supply of `n workers, who are each endowed with one unit of labor that is supplied inelastically.
2.1 Preferences
The representative consumer in country n has the following homothetic indirect utility function:
un =wnP (pn)
, (1)
where pn is the vector of prices in country n of the goods produced by each country i with elements pni (inclusive of
trade costs); wn is the wage; and P (·) is a continuous and twice di�erentiable function that corresponds to the ideal
price index for consumption. From Roy’s Identity, country n’s demand for the good produced by country i is:
cni = cni (pn) = −∂ (1/P (pn))
∂pniwnP (pn) . (2)
2.2 Production
Each country’s good is produced with labor according to a constant returns to scale production technology, with
productivity zi in country i. Markets are perfectly competitive. Goods can be traded between countries subject to
iceberg trade costs, such that τni ≥ 1 units of a good must be shipped from country i in order for one unit to arrive
in country n (where τni > 1 for n 6= i and τnn = 1). Therefore, the cost in country n of consuming one unit of the
good produced by country i is:
pni =τniwizi
. (3)
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2.3 Expenditure Shares and Market Clearing
Country n’s expenditure share on the good produced by country i can be written as:
sni =pnicni (pn)∑N`=1 pn`cn` (pn)
. (4)
Totally di�erentiating this expenditure share equation, the proportional change in expenditure shares in country n
depends on the proportional change in the prices of the goods from each country i and the own and cross-price
elasticities for each good:
d ln sni =N∑h=1
[θnih −
N∑k=1
snkθnkh
]d ln pnh, (5)
where
θnih ≡(∂ (pnicni (pn))
∂pnh
pnhpnicni (pn)
),
is the elasticity in country n of the expenditure share for good i with respect to the price of good h. Totally di�eren-
tiating prices, the proportional change in the price in country n of the good produced by country i depends on the
proportional changes in the underlying trade costs, wages and productivities as follows:
d ln pni = d ln τni + d lnwi − d ln zi. (6)
Market clearing requires that income in country i equals the expenditure on goods produced by that country:
wi`i =N∑n=1
sniwn`n, (7)
where for simplicity we begin by considering the case of balanced trade and show how the analysis generalizes to
imbalanced trade in Section 4 below.
2.4 Comparative Statics
Using preferences (1) and market clearing (7), we now characterize the general equilibrium e�ect of shocks to pro-
ductivity and trade costs. First, totally di�erentiating the market clearing condition (7) holding constant country
endowments, the change in income in each country i depends on the share of value-added that it derives from each
market n (tin), the own and cross-price elasticities (θnih), and the proportional changes in the price of the good from
each country h as determined by (6):
d lnwi =N∑n=1
tin
(d lnwn +
[N∑h=1
[θnih −
N∑k=1
snkθnkh
][ d ln τnh + d lnwh − d ln zh]
]), (8)
where the share of value-added that country i derives from each market n is de�ned as:
tin ≡sniwn`nwi`i
. (9)
Second, totally di�erentiating the indirect utility function (1), the change in welfare in country n equals the change
in income in that country minus the expenditure share weighted average of the proportional change in the price of
each country’s good, as determined by (6):
d lnun = d lnwn −N∑i=1
sni [ d ln τni + d lnwi − d ln zi] . (10)
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The market clearing condition for each country (8) shapes how exogenous changes in productivities ( d ln zi) and
trade costs ( d ln τni) map into endogenous changes in wages ( d lnwi). The utility function (10) determines how
these endogenous changes in wages ( d lnwi) and the exogenous changes in productivities ( d ln zi) and trade costs
( d ln τni) translate into endogenous changes in welfare in each country ( d lnun). In general, both the own and cross-
price elasticities of expenditure with respect to prices (θnih) are variable and depend on the entire price vector (pn),
complicating the mapping from exogenous to endogenous variables.
3 Constant Elasticity of Import Demand
We now show that a sharp “friends” and “enemies” representation of countries’ income and welfare exposure to foreign
productivity or trade cost shocks can be obtained under the assumption of a constant trade elasticity. In Subsections
3.2 through 3.5, we derive this representation for small changes in productivity or trade costs under this assumption
of a constant trade elasticity. In Subsection 3.6, we characterize the quality of the approximation for large changes as
a function of the properties of the observed trade matrices, and show this approximation to be almost exact even for
productivity shocks of the magnitude implied by the observed trade data.
Throughout this section, we derive our results in a single-sector, constant elasticity Armington model, which is a
special case of the framework developed in the previous section. In Section E of the online appendix, we show that
these results hold in the entire class of international trade models considered in Arkolakis et al. (2012), henceforth ACR,
which satisfy the four primitive assumptions of (i) Dixit-Stiglitz preferences; (ii) one factor of production; (iii) linear
cost functions; and (iv) perfect or monopolistic competition; as well as the three macro restrictions of (i) a constant
elasticity import demand system, (ii) a constant share of pro�ts in income, and (iii) balanced trade. In addition to
the Armington model considered here, this class includes models of perfect competition and constant returns to scale
with Ricardian technology di�erences, as in Eaton and Kortum (2002), and those of monopolistic competition and
increasing returns to scale, in which goods are di�erentiated by �rm, as in Krugman (1980) and Melitz (2003) with an
untruncated Pareto productivity distribution.
While at the beginning of this section we allow for both productivity and trade cost shocks, we focus from sub-
section 3.3 onwards on productivity shocks alone. In Section 4 below, we consider a variety of extensions, including
trade imbalances, trade cost shocks, multiple sectors following Costinot et al. (2012), input-output linkages following
Caliendo and Parro (2015), and economic geography. We also derive sensitivity bounds for countries’ income and
welfare exposure to foreign productivity shocks for the more general case of a variable trade elasticity.
3.1 Trade Matrices
We begin by de�ning some notation. We use boldface, lowercase letters for vectors, and boldface, uppercase letters
for matrices. We use the corresponding non-bold, lowercase letters for elements of vectors and matrices. We use I
to denote the N × N identity matrix. We now introduce two key matrices of trade shares that we show below are
central to determining the impact of productivity and trade cost shocks.
Expenditure Share and Income Share Matrices Let S be the N × N matrix with the ni-th element equal to
importer n’s expenditure on exporter i. Let T be the N × N matrix with the in-th element equal to the fraction of
income that exporter i derives from selling to importer n. We refer to S as the expenditure share matrix and to T as
7
the income share matrix. Intuitively, sni captures the importance of i as a supplier to country n, and tin captures the
importance of n as a buyer for country i. Note the order of subscripts: in matrix S, rows are buyers and columns are
suppliers, whereas in matrix T, rows are suppliers and columns are buyers. Both matrices have rows that sum to one.
These S and T matrices are equilibrium objects that can be obtained directly from observed trade data. We derive
comparative statics results using these observed matrices. Using Sk to represent the matrix S raised to the k-th power,
we impose the following technical assumption on the matrix S, which is satis�ed in the observed trade �ow data.
Assumption 1. (i) For any i, n, there exists k such that[Sk]in> 0. (ii) For all i, sii > 0.
The �rst part of this assumption states that all countries trade with each other directly or indirectly. That is, in
the language of graph theory, the global trade network is strongly connected. This assumption is important because
shocks propagate in general equilibrium through changes in relative prices, which are only well-de�ned if countries
are connected (potentially indirectly) to each other through trade. When the global trade network has disconnected
components—for instance, if a subset of countries only trade among themselves but not with other nations, or if some
countries are in autarky—our results can be applied to study the general equilibrium propagation of shocks within
each of the connected components separately. In practice, we �nd that the global trade network is strongly connected
throughout our sample period. The second part of this assumption ensures that every country consumes a positive
amount of domestic goods, which again is satis�ed in all years.
Using Assumption 1, we now establish a relationship between the S and T matrices, which shapes the general
equilibrium impact of productivity shocks on income and welfare.
Lemma 1. Assuming that trade is balanced,
1. S has a unique left-eigenvector q′ with all positive entries summing to one; the corresponding eigenvalue is one.
2. The i-th element of this left-eigenvector qi is the equilibrium income of country i relative to world nominal GDP,
qi = wi`i/(∑N
n=1 wn`n
).
3. q′ is also a left-eigenvector of T with eigenvalue one, and qitin = qnsni.
4. Under free-trade (i.e. τni = 1 for all n, i), q′ is equal to every row of S and of T.
Proof. See Section B.1 of the online appendix.
Going forward, we refer to the vector q′ as simply the income vector, re�ecting our normalization that world
nominal GDP is equal to one. Lemma 1 shows that, under balanced trade, one could recover q and T from the
expenditure share matrix S. A key implication of this result is that S is a su�cient statistic for the general equilibrium
e�ect of small productivity shocks on income and welfare under balanced trade.2
In the remainder of this section, we use these properties of the trade matrices to characterize the �rst-order general
equilibrium e�ects of global productivity shocks on income and welfare in each country in the constant elasticity
version of the Armington model developed in Section 2 above.2As the expenditure and income shares sum to one, both the matricesS andT represent row-stochastic Markov chains, andq′ is their stationary
distribution. Assumption 1 ensures that the matrix S is primitive. Since the elements of the matrix T satisfy qitin = qnsni, the Markov chain Sis reversible if and only if S = T, which holds if and only if trade is balanced bilaterally between each country-partner-pair, a condition which isnot satis�ed in the data. Finally, the matrix TS, which we show below determines the cross-price elasticity under a constant trade elasticity, is themultiplicative reversiblization of S (Fill 1991), with qi [TS]in = qn [TS]ni. Note that the income vector q′ is a left-eigenvector of this matrix TSwith eigenvalue one.
8
3.2 First-Order Comparative Statics
In the constant elasticity Armington speci�cation, the preferences of the representative consumer in country n in
equation (1) are characterized by the following functional form:
un =wn[∑N
i=1 p−θni
]− 1θ
, θ = σ − 1, σ > 1, (11)
where σ > 1 is the constant elasticity of substitution between country varieties and θ = σ − 1 is the trade elasticity.
Using Roy’s Identity, country n’s share of expenditure on the good produced by country i is:
sni =p−θni∑N
m=1 p−θnm
. (12)
Using these functional forms in the market clearing condition (7) and totally di�erentiating holding constant
country endowments, the system of equations for the change in income (8) now simpli�es to:
d lnwi =
N∑n=1
tin
(d lnwn + θ
(N∑h=1
snh [ d ln τnh + d lnwh − d ln zh]− [ d ln τni + d lnwi − d ln zi]
)). (13)
The system of equations for the change in welfare again takes the same form as in equation (10):
d lnun = d lnwn −N∑i=1
sni [ d ln τni + d lnwi − d ln zi] . (14)
Given exogenous changes in productivities ( d ln zi) and trade costs ( d ln τni), the market clearing condition for each
country (8) provides a system of N equations that can be used to determine the N endogenous changes in wages
in each country ( d lnwi). Combining these endogenous changes in wages ( d lnwi) with the exogenous changes in
productivities ( d ln zi) and trade costs ( d ln τni), the utility function (10) determines the N endogenous changes in
welfare in each country ( d lnun).
3.3 Su�cient Statistics for Income and Welfare Exposure to Productivity Shocks
We now use these comparative statics results in equations (13) and (14) to obtain our “friend” and “enemy” su�cient
statistics for countries’ income and welfare exposure to foreign shocks. To streamline the exposition and in light of
our empirical application, we now focus on productivity shocks ( d ln zi 6= 0), assuming that trade cost shocks are zero
( d ln τni = 0 ∀n, i). In Subsection 4.1 in the next section, we show that our approach naturally also accommodates
trade cost shocks ( d ln τni 6= 0).
3.3.1 Su�cient Statistics for Income Levels
We begin by showing that the �rst-order general equilibrium e�ects of small productivity shocks in each country
on income in all countries in equation (13) have a matrix representation, which has two key advantages for our
purposes. First, we can use this representation to recover our “friend” and “enemy” measures of countries’ exposure
to a foreign productivity shock as a simple matrix inversion problem, which can be solved almost instantaneously
to machine precision. Second, this representation isolates key mechanisms in the model that enable us to relate its
quantitative predictions to underlying economic forces. Using d ln z and d ln w to denote column vectors of country-
level productivity shocks and wage responses, we have the following matrix representation of equation (13).
9
Proposition 1. Under ACR assumptions (i)-(iv) and macro restrictions (i)-(iii), the �rst-order general equilibrium impact
of productivity shocks on income in all countries around the world solves the �xed point equation:
d ln w︸ ︷︷ ︸income e�ect
= T d ln w︸ ︷︷ ︸market-size e�ect
+ θ ·M× ( d ln w − d ln z)︸ ︷︷ ︸cross-substitution e�ect
, (15)
where M ≡ TS− I is an N ×N matrix with in-th entrymin ≡∑Nh=1 tihshn − 1n=i.
Proof. The proposition follows immediately from equation (13) and our assumption that that d ln τni = 0 ∀n, i, as
shown in Section D of the online appendix.
From equation (15), we can compute the e�ect of small productivity shocks on income in each country around the
world using the income share matrix (T) and the cross-substitution matrix (M), both of which are transformations of
the expenditure share matrix (S). The matrix T in the �rst term on the right-hand side captures a market-size e�ect:
To the extent that the productivity shock vector d ln z increases incomes in countries n, this raises income in country
i through increased demand for its goods. In particular, the elements of T are the share of income that country i earns
through selling to each market n (tin), and capture how dependent country i is on markets in each country n.
The matrix M in the second term on the right-hand side captures a cross-substitution e�ect. To understand this
e�ect, consider the in-th element of this matrix: min ≡∑Nh=1 tihshn−1n=i. For i 6= n, the sum
∑Nh=1 tihshn captures
the overall competitive exposure of country i to country n, through each of their common markets h, weighted by
the importance of market h for country i’s income (tih). As the competitiveness of country n increases, as measured
by a decline in its wage relative to its productivity ( d lnwn− d ln zn), consumers in all markets h substitute towards
countryn and away from other countries i 6= n, thereby reducing income in country i and raising it in countryn. With
a constant elasticity import demand system, the magnitude of this cross-substitution e�ect in market h depends on
the trade elasticity (θ) and the share of expenditure in market h on the goods produced by country n (shn): consumers
in market h increase expenditure on country n by (shn − 1) and lower expenditure on country i by shn. Summing
across all markets h, we obtain the overall impact of the shock to country n’s production cost on country i’s income,
as captured in the in-th element of the matrix M.
We now use this matrix representation in Proposition 1 to recover our “friend” and “enemy” measures of coun-
tries’ bilateral income exposure to productivity growth. As the trade share matrices T and M in equation (15) are
homogenous of degree zero in incomes, they do not pin down the level of changes in nominal incomes. As in any
general equilibrium model, we need a choice of numeraire. We choose world GDP as our numeraire, which with un-
changed country endowments (`i) implies the following normalization:∑Ni=1 qi d lnwi = 0. Starting with equation
(15), dividing both sides by (θ + 1), re-arranging terms, and using this normalization, we obtain:
(I−V) d lnw = − θ
θ + 1M d ln z, V ≡ T + θTS
θ + 1−Q, (16)
where Q is an N ×N matrix with the income row vector q′ stacked N times and recall that we assume θ > 0. Under
free-trade (i.e. τni = 0 for all n, i), Q = S = T.
The presence of the term Q d ln w = 0 on the left-hand side in equation (16) re�ects our choice of numeraire. In the
absence of this term, the matrix(I− T+θTS
θ+1
)is not invertible: the income shares and expenditure shares sum to one
(∑Nn=1 tin = 1 and
∑Nn=1 sni = 1), thus the rows of T+θTS
θ+1 also sum to one, and the columns of(I− T+θTS
θ+1
)are
10
not linearly independent. This non-invertibility re�ects the fact that income can only be recovered from expenditure
shares up to a normalization or choice of units. While we choose world GDP as our numeraire because it is convenient
for the matrix inversion,3 all of our predictions for relative country incomes are invariant to whatever normalization
is chosen. Using equation (16), we are now in a position to formally state the following de�nition.
De�nition 1. Our friends-and-enemies matrix for income is de�ned as:
W ≡ − θ
θ + 1(I−V)
−1M. (17)
From De�nition 1 and equation (16), our friends-and-enemies matrix W completely summarizes the general equi-
librium e�ect of small productivity shocks on income in each country around the world.
Corollary 1. Income exposure to global productivity shocks is:
d ln w = W d ln z (18)
Proof. The corollary follows immediately from Proposition 1, De�nition 1, and our choice of world GDP as numeraire
(Q d ln w = 0), as shown in Section D of the online appendix.
The elements of this matrix W capture countries’ bilateral income exposure to productivity shocks. In particular,
the in-th element of this matrix is the elasticity of income in country i (row) with respect to a small productivity shock
in country n (column). We refer to country n as being a “friend” of country i for income when this elasticity is positive
and an “enemy” of country i for income when this elasticity is negative. In general, W is not necessarily symmetric:
i could view n as a friend, while n views i as an enemy. Finally, we now establish that the friends-and-enemies matrix
W in De�nition 1 exists, because the matrix (I − V ) is invertible under Assumption 1.
Lemma 2. Let V ≡ T+θTSθ+1 − Q. Under Assumption 1 and θ > 0, the matrix (I−V) is invertible, (I − V)−1 =∑∞
k=0 Vk , and the power series converge at rate |µ| < 1, where |µ| is the absolute value of the largest eigenvalue of V
(i.e., ||Vk|| ≤ c · |µ|k for some constant c).
Proof. See Section B.2 of the online appendix.
Therefore, given the observed trade matrices (S, T and M), we obtain a complete characterization of the general
equilibrium e�ect of small productivity shocks under our assumption of a constant trade elasticity.
3.4 Su�cient Statistics for Welfare
We next show that the general equilibrium e�ects of small productivity shocks in all countries on welfare in each
country in equation (14) have an analogous matrix representation, which again allows us to connect quantitative
predictions directly to underlying economic mechanisms in the model. Using d ln u to denote the column vector of
country-level welfare changes, we have the following matrix representation of equation (14).3Note that the matrix T+θTS
θ+1represents a row-stochastic Markov chain; its left eigenvector q′ is also the stationary distribution of the Markov
chain, and limk→∞(
T+θTSθ+1
)k= Q.
11
Proposition 2. Under ACR assumptions (i)-(iv) and macro restrictions (i)-(iii), the �rst-order general equilibrium impact
of productivity shocks on welfare in all countries around the world solves the following �xed point equation:
d ln u︸ ︷︷ ︸welfare e�ect
= d ln w︸ ︷︷ ︸income e�ect
− S ( d ln w − d ln z)︸ ︷︷ ︸cost-of-living e�ect
. (19)
Proof. The proposition follows immediately from equation (14) and our assumption that d ln τni = 0 ∀n, i, as shown
in Section D of the online appendix.
From Propositions 1 and 2, we can compute the e�ect of productivity shocks on the welfare of all countries around
the world using the income share matrix (T), the cross-substitution matrix (M), and the expenditure share matrix (S),
where both the income share and cross-substitution matrices are transformations of the expenditure share matrix.
The presence of this expenditure share matrix (S) in the second term on the right-hand side of equation (19) captures
a cost of living e�ect, which re�ects the impact of the productivity shock in country i on the price index in country
n. The elements of this matrix sni capture the relative importance of each country i in the consumer expenditure
bundle of country n. A productivity shock in country i will have a large positive e�ect on welfare in country n if it
has a large positive e�ect on wages in country n (through the income e�ect) and a large negative e�ect on wages and
production costs in the countries from which country n sources most of its goods (through the cost of living e�ect).
As the elements of the expenditure share matrix (S) are homogeneous of degree zero in per capita income and sum
to one for each importer, adding any constant vector c to changes in log per capita incomes ( d ln w = d ln w + c)
leaves the welfare e�ect in equation (19) unchanged (since c− Sc = 0). Therefore, the welfare e�ect in Proposition
2 is invariant to our choice of numeraire. Using Corollary 1 and Proposition 2, we are now in a position to formally
state the following de�nition.
De�nition 2. Our friends-and-enemies matrix for welfare is de�ned as:
U ≡ (I− S) W + S. (20)
From De�nition 2, our friends-and-enemies matrix U completely summarizes the general equilibrium e�ect of small
productivity shocks on welfare in each country around the world.
Corollary 2. Welfare exposure to global productivity shocks is:
d ln u = U d ln z. (21)
Proof. The corollary follows immediately from Corollary 1, De�nition 2 and Proposition 2.
The elements of this matrix U capture countries’ bilateral welfare exposure to productivity shocks. In particular,
the ni-th element of this matrix is the elasticity of welfare in country n (row) with respect to a small productivity
shock in country i (column). We refer to country i as being a “friend” of country n for welfare when this elasticity
is positive and an “enemy” of country n for welfare when this elasticity is negative. As for income exposure, welfare
exposure U is not necessarily symmetric: i could view n as a friend, while n views i as an enemy.
12
3.5 Economic Mechanisms
We now use our friends-and-enemies matrix representation to isolate the key economic mechanisms through which
a productivity shock in one country a�ects income and welfare in all countries in this class of international trade
models with a constant trade elasticity.
1. Partial and General Equilibrium E�ects Our measure of the overall impact of foreign productivity shocks
on domestic income in equation (17) includes both the direct (partial equilibrium) e�ect of these productivity shocks
( d ln z) on competitiveness in each market, as well as their indirect (general equilibrium) e�ects on competitiveness
and the size of each market through endogenous changes in incomes ( d ln w). To separate these two e�ects, we use
the property that the spectral radius of V is less than one, which allows us to re-write our income exposure measure
as the following power series:
W = − θ
θ + 1(I−V)
−1M = − θ
θ + 1
∞∑k=0
VkM = − θ
θ + 1M︸ ︷︷ ︸
partial equilibrium
− θ
θ + 1
(V + V2 + . . .
)M︸ ︷︷ ︸
general equilibrium
, (22)
where the �rst term on the right-hand side (− θθ+1M ) captures the partial equilibrium e�ect (the direct e�ect of higher
productivity in country ` on income in country i, evaluated at the initial values of incomes in each country); the second
term on the right-hand side (− θθ+1VM) and the following higher-order terms in V capture the general equilibrium
e�ect (through endogenous changes in incomes in each country).4
2. Market-Size and Cross-Substitution E�ects From Proposition 1, overall income exposure to productivity
shocks is jointly determined by the market-size and cross-substitution e�ects. To separate out the contributions of
each of these mechanisms to general equilibrium changes in incomes, we undertake a counterfactual exercise in
which we impose that the market-size e�ect is the same for all countries and allow only the cross-substitution e�ect
to di�er across countries. Speci�cally, we replace the term T d ln w in equation (15) with Q d ln w, so that the general
equilibrium income response to productivity shocks d ln wSub solves the �xed point equation:
d ln wSub = Q d ln wSub + θ ·M(
d ln wSub − d ln z), (23)
where we use the superscript Sub to indicate cross-substitution e�ect.
In our actual income exposure measure in equation (15), the rows of the matrix T vary across countries iwith the
shares of markets in their income (t′). In contrast, in this counterfactual income exposure measure in equation (23),
the rows of the matrix Q are the same across countries i and equal to the shares of markets in world income (q′).
Using our choice of world GDP as numeraire (Q d ln w = 0), we can recover counterfactual income exposure from
the cross-substitution e�ect alone from the following matrix inversion:
WSub ≡ −θ (I− θ (M + Q))−1
M. (24)
d ln wSub = WSub d ln z. (25)4Here we de�ne the direct or partial equilibrium e�ect for a given country as holding all other countries incomes constant at their values in the
initial equilibrium before the productivity shock, but other de�nitions are possible, as discussed in a di�erent context in Huo et al. (2019).
13
While our friends-and-enemies matrix for income W from the previous subsection captures overall income exposure
to productivity shocks; the matrix WSub ≡ −θM (I− θ (M + Q))−1 captures income exposure through the cross-
substitution e�ect alone; and the di�erence W −WSub captures income exposure through the market-size e�ect.
3. Contribution of Third Markets to Bilateral Income Exposure Our measures of exposure to productivity
shocks capture all mechanisms through which productivity shocks a�ect income and welfare in the model, including
both imports and exports and both own and third-market e�ects. We now use our approach to evaluate how much of
one country’s exposure to productivity shocks operates through third markets. Let G denote the subset of countries
for which we are interested in third-market e�ects (e.g. for U.S. income exposure to a Chinese productivity shocks, G
might be the European Union). To evaluate the contribution of these third markets to income and welfare exposure,
we construct counterfactual expenditure share matrices excluding them.
In particular, we de�ne S−G as the transformed expenditure share matrix, removing the k-th rows and columns
from S for all k ∈ G, and rescaling the remaining rows to sum to one. Using this counterfactual expenditure share
matrix S−G , we construct the corresponding income share matrix T−G and cross-substitution matrix M−G . Using
these counterfactual trade share matrices, we recompute both our overall measure of income exposure (W−G ) using
equation (17) and the cross-substitution e�ect (WSub−G ) using equation (24). Comparing these measures to those includ-
ing all countries (W, WSub), we can quantify the importance of this group of third markets for both overall income
exposure and the cross-substitution e�ect.
Finally, welfare exposure (U) in De�nition 2 is a linear combination of income exposure (W) and the expenditure
share matrix (S) that controls the cost of living e�ect. Therefore, substituting each of the above decompositions of
income exposure (W) into welfare exposure (U), we can quantify the contribution of each of these mechanisms to
the impact of productivity shocks on welfare.
3.6 Comparison with Exact Hat-Algebra
Our friends-and-enemies exposure measures have the advantage that they are quick and easy to compute using only
matrices of observed trade data. They also allow researchers working with quantitative trade models to transparently
assess the role of di�erent economic mechanisms. A potential limitation is that our exposure measures correspond
to �rst-order e�ects in a linearization that is only exact for small changes, which raises the question of how good
an approximation they provide for large changes. We now characterize the quality of this approximation by relating
the magnitude of the second and higher-order terms in the Taylor-series expansion to properties of the observed
trade matrices. In our later empirical analysis, we use these results to show that our linearization is almost exact for
productivity shocks, even for large changes of the magnitude implied by the observed trade data.
We begin by comparing our linearization to the full non-linear solution of the model for large changes using the
exact-hat algebra approach of Dekle et al. (2007). In particular, using this exact-hat algebra approach, we can re-write
the market clearing condition (7) in a counterfactual equilibrium following a productivity shock (denoted by a prime)
in terms of the observed values of variables in an initial equilibrium (no prime) and the relative changes of variables
between the counterfactual and initial equilibria (denoted by a hat such, that x = x′/x):
ln wi =
(θ
θ + 1
)ln zi +
1
θ + 1ln
[N∑n=1
tinwn∑N
`=1 sn`w−θ` zθ`
], (26)
14
which provides a system of N equations that can be solved for the N unknown relative changes in wages (wn) given
the assumed productivity shock (z`) and the observed trade shares (tin, sni) in the initial equilibrium.
Using equation (15), we can re-write our friends-and-enemies income exposure measure in the following similar
but log linear form:
d lnwi =
(θ
θ + 1
)d ln zi +
1
θ + 1
N∑n=1
tin
[d lnwn + θ
N∑`=1
sn` [ d lnw` − d ln z`]
]. (27)
Comparing equations (26) and (27), we �nd that the di�erence between the predictions of the exact-hat algebra
and our friend-enemy linearization corresponds to the di�erence between the log of a weighted mean and a weighted
mean of logs. These two expressions take the same value as trade costs become large (tnn → 1, snn → 1 for all
n) and under free trade. More generally, these two expressions take di�erent values, with the di�erence between
them equal to the second and higher-order terms in a Taylor-series expansion. We now characterize the properties
of the second-order term in this expansion, before bounding the magnitude of all higher-order terms. To simplify
notation, we de�ne zi as ln zi. We use fi (z) to denote the implicit function that de�nes the log changes in wages wiin equation (26) as a function of the log productivity shocks {z}, and we use εi (z) to denote the second-order term
in the Taylor-series expansion of fi (z). Using this notation, we can rewrite equation (26) as:
wi = −θ (wi − zi) +∑n
tinwn + θ∑n
min [wn − zn]︸ ︷︷ ︸�rst-order
+ εi (z)︸ ︷︷ ︸second-order
+O(‖z‖3
)︸ ︷︷ ︸higher-order
.
The properties of the second-order term depend on the Hessian Hfi of the function fi evaluated at z` = 0 ∀ `:
Hfi ≡
∂2fi(0)∂z21
∂2fi(0)∂z1∂z2
· · · ∂2fi(0)∂z1∂zN
∂2fi(0)∂z2∂z1
∂2fi(0)∂z22
· · · ∂2fi(0)∂z2∂zN
......
. . ....
∂2fi(0)∂zN∂z1
∂2fi(0)∂zN∂z2
· · · ∂2fi(0)∂z2N
, (28)
where we can write this second-order term as εi (z) = z′Hfi z.
We now proceed as follows. First, we derive an expression for this Hessian in terms of matrices of observed
trade data (Proposition 3). Second, we show that a cross-country average of the second-order terms is exactly zero
(Proposition 4). Third, we show that the absolute magnitude of this second-order term for each country can be bounded
by the largest eigenvalue (in absolute) value of this Hessian (Proposition 5). As this largest eigenvalue can be measured
using observed trade data, we can use this result to bound the quality of the approximation for each country given the
observed trade matrices. Fourth, we aggregate these results for the second-order terms across countries, and provide
an upper bound on their sums of squares (Proposition 6). Again this bound can be computed using observed trade
data and provides a summary measure of the overall performance of our linearization. Finally, Proposition 7 provides
a bound on all higher order terms, including the second-order term and beyond.
In Proposition 3, we show that the Hessian (Hfi ) depends solely on the trade elasticity (θ) and the three observed
matrices that capture the market-size e�ects (T), cross-substitution e�ects (M), and expenditure shares (S). In par-
ticular, the second-order term depends on expectations and variances taken across the elements of these matrices, as
summarized in the following proposition.
15
Proposition 3. The Hessian matrix can be explicitly written as
Hfi = −1
2(A′ (diag ([M + I]i)− S′diag (Ti) S) A−B′ (diag (Ti)−T′iTi) B) .
where A ≡ θθ+1 (I−V)
−1(I−T) and B ≡ θ
θ+1 (I−V)−1
M + SA, and Ti, Mi are the i-th rows of T and M,
respectively.
The second-order term εi (z) ≡ z′Hfi z can be re-written more intuitively as
εi (z) = −θ2ETiVSn [ln wk − zk]
2+
VTi (ln wi + θESn [ln wk − zk])
2,
where ETi , EMi, ESn , VTi , and VSn are expectations and variances taken using {Tin}Nn=1, {Min}Nn=1, and {Snk}
Nk=1
as measures (e.g. ETi [xn] ≡∑Nn=1 Tinxn, VTi [xn] ≡
∑Nn=1 Tinx
2n −
(∑Nn=1 Tinxn
)2
).
Proof. See Section B.3 of the online appendix.
As a �rst step towards characterizing the magnitude of the second-order terms in this expression, we next show
in Proposition 4 that the average across countries (weighted by country size in the initial equilibrium before the
productivity shock) of these second-order terms is exactly zero: q′ε (z) = 0. Therefore, these second-order terms
raise or reduce the predicted change in the wage of individual countries in response to the productivity shock, but
when weighted appropriately they average out across countries.
Proposition 4. Weighted by each country’s income, the second-order terms average to zero for any productivity shock
vector: q′ε (z) = 0 for all z.
Proof. See Section B.4 of the online appendix.
We now bound the absolute value of the second-order term for the income response of each country, following any
vector of productivity shocks. First, note that because the model features constant returns to scale, a uniform shock to
the productivity of all countries across the globe does not a�ect relative income. It is therefore without loss of gener-
ality to focus on productivity shocks that average to zero. We now show in Proposition 5 that the absolute value of the
second-order term for the log-change in income of each country i is bounded, relative to the variance of productivity
shocks, by the largest eigenvalue µmax,i (by absolute value) of the Hessian matrix Hfi (|εi (z)| ≤∣∣µmax,i
∣∣ · zT z).
The corresponding eigenvector zmax,i is the productivity shock vector that achieves the largest second-order term
for country i. As these eigenvalues of the Hessian matrix for each country can be evaluated using the observed trade
matrices, we thus obtain a bound on the size of second-order term for each country that can be computed in practice
using the observed trade data. In our empirical application below, we show that for each country, even the largest
eigenvalue is close to zero, which in turn implies that the second-order term for each country is close to zero.
Proposition 5. |εi (z)| ≤∣∣µmax,i
∣∣ · z′ z for all z, where µmax,i is the largest eigenvalue of Hfi by absolute value.
Let zmax,i denote the corresponding eigenvector (such that Hfi zmax,i = µmax,izmax,i). The upper bound for |εi (z)| is
achieved when productivity shocks are represented by zmax,i:∣∣εi (zmax,i
)∣∣ =∣∣µmax,i
∣∣ · (zmax,i)T
zmax,i.
Proof. See Section B.5 of the online appendix.
16
We next aggregate the second-order terms across countries and provide an upper-bound on their sum-of-squares
in Proposition 6, which enables us to assess the overall performance of our linear approximation. As we show in our
empirical application later, the standard unit vector e` comes close to achieving the upper-bound for the `-th equation,
i.e. e` ≈ zmax,` for all `. Intuitively, because ei is orthogonal to ej for all i 6= j, this implies that the productivity
shock vectors zmax,i and zmax,j that maximize second-order e�ects for di�erent countries i 6= j are almost orthogonal.
Hence, given any productivity shock vector z, at most one country ln wi = fi (z) can have a second-order term close
to the upper-bound µmax,i, which is small, and the second-order terms for all other countries are close to zero. To
formalize this intuition, Proposition 6 constructs a symmetric order-4-tensor A such that√
1N
∑Ni=1 ε
2i (z)
zT zis bounded
above by the square-root of the spectral norm of A. Note that 1N
∑Ni=1 ε
2i (z) is exactly the mean-square-residuals
from a linear regression of the second-order-approximation on our linearized solution.
Proposition 6. Let A : RN → R≥0 denote the order-4 symmetric tensor de�ned by the polynomial
g (z) =1
N
N∑i=1
N∑a,b,c,d=1
[Hfi ]ab · [Hfi ]cd · za · zb · zc · zd
,
where [Hfi ]ab is the ab-th entry of Hfi . By construction, g (z) = 〈A, z⊗ z⊗ z⊗ z〉 represents the inner product and is
equal to the cross-equation sum-of-square of the second-order terms (g (z) = 1N
∑i ε
2i (z)) under productivity shock z.
Let µA be the spectral norm of A:
µA ≡ supz
〈A, z⊗ z⊗ z⊗ z〉‖z‖42
,
where ‖ · ‖2 is the `2 norm (‖z‖2 ≡√
z′z). Then√1
N
∑i
ε2i (z) ≤√µA‖z‖22 =
õAz
′z.
Proof. See Section B.6 of the online appendix.
The spectral norm of A can be computed using the observed trade data, and the norm being close to zero implies
that the second-order terms are close to zero. Furthermore, Lagrange’s remainder theorem implies that if productivity
shocks are bounded, we can obtain a bound on all the higher-order terms including second-order and above. Using
Hfi (z) to denote the Hessian of fi (z) evaluated at productivity shock z (not necessarily equal to the zero vector),
we have the following result.
Proposition 7. Suppose productivity shocks are bounded, z ∈ X ≡∏Ni=1 [z, z]. For any z, there exists x ∈ X such that
ln wi = −θ (ln wi − zi) +∑n
tin ln wn + θ∑n
min [ln wn − zn]︸ ︷︷ ︸�rst-order
+ z′Hfi (x) z︸ ︷︷ ︸second andhigher-order
.
Proof. This is a direct application of Lagrange’s remainder theorem.
Proposition 7 demonstrates that the Hessian matrix, evaluated at some productivity shock vector x, provides the
exact error for our �rst-order approximation. A bound on the eigenvalue of the Hessian evaluated over the entire
support X of productivity shocks therefore provides an upper-bound on the exact approximation error. We exploit
this result in our empirical analysis below and show that approximation errors are close to zero for productivity shocks
17
of the magnitude implied by the observed trade data. We thus conclude that our linearization provides an almost exact
approximation to the full non-linear solution of the model given the observed trade matrices. Consistent with this,
when we regress the full non-linear solution from the exact-hat algebra on our linear approximation in our empirical
analysis below, we �nd R-squared close to one (R2 > 0.99) in all of our simulations.
4 Extensions
We now consider a number of extensions to our friends-and-enemies measures of countries’ income and welfare
exposure to productivity shocks. In Section 4.1, we derive the corresponding matrix representations allowing for
both productivity and trade cost shocks. In Section 4.2, we relax one of the ACR macro restrictions to allow for trade
imbalance. In Section 4.3, we relax another of the ACR macro restrictions to consider small deviations from a constant
elasticity import demand system. In Section 4.4, we show that our results generalize to a multi-sector model with a
single constant trade elasticity following Costinot et al. (2012). In Section 4.5, we extend this speci�cation further to
introduce input-output linkages following Caliendo and Parro (2015). Finally, in Section 4.6, we show that our results
also hold for economic geography models with factor mobility, including Helpman (1998), Redding and Sturm (2008),
Allen and Arkolakis (2014), Ramondo et al. (2016) and Redding (2016).
4.1 Productivity and Trade Cost Shocks
Whereas productivity shocks are common across all trade partners, trade cost shocks are bilateral, which implies that
our comparative static results in equations (13) and (14) now have a representation as a three tensor. To reduce this
three tensor down to a matrix (two tensor) representation, we aggregate bilateral trade costs across partners using
the appropriate weights implied by the model. In particular, we de�ne two measures of outgoing and incoming trade
costs, which are trade-share weighted averages of the bilateral trade costs across all export destination and import
sources, respectively. We de�ne outgoing trade costs for country i as d ln τouti ≡∑n tin d ln τni, where the weights
are the income share (tin) that country i derives from selling to each export destination n. We de�ne incoming trade
costs for country n as d ln τ inn ≡∑i sni d ln τni, where the weights are the expenditure share (sni) that country n
devotes to each import source i. Using these de�nitions in equations (13) and (14), we obtain:
d lnwi =N∑n=1
tin d lnwn + θ
( ∑Nh=1
∑Nn=1 tinλnh [ d lnwn − d ln zn]− [ d lnwi − d ln zi]
+∑Nn=1 tin d ln τ inn − d ln τouti
), (29)
d lnun = d lnwn −N∑i=1
sni [ d lnwi − d ln zi]− d ln τ in, (30)
which enables us to obtain the following matrix representation.
Proposition 8. Under ACR assumptions (i)-(iv) and macro restrictions (i)-(iii), the �rst-order general equilibrium impact
of productivity and trade cost shocks on income and welfare in all countries around the world solves the following �xed
point equations:
d ln w︸ ︷︷ ︸income e�ect
= T d ln w︸ ︷︷ ︸market-size e�ect
+ θ[M ( d ln w − d ln z) + T d ln τ in − d ln τout
]︸ ︷︷ ︸cross-substitution e�ect
(31)
= W d ln z + θ (I−V)−1 (
T d ln τ in − d ln τout)
18
d ln u︸ ︷︷ ︸welfare e�ect
= d ln w︸ ︷︷ ︸income e�ect
−S ( d ln w − d ln z) + d ln τ in︸ ︷︷ ︸cost-of-living e�ect
(32)
Proof. The proposition follows from equations (29) and (30), as shown in Section F.1 of the online appendix.
From Proposition 8, holding productivity constant, country n’s demand for the goods supplied by country i in-
creases if the bilateral trade cost τni between these countries falls relative to country n’s trade costs with all other
nations. These e�ects are aggregated into d ln τ inn and d ln τouti , which weight the bilateral changes in trade costs
by their appropriate income and expenditure shares. From equation (31), country i’s income increases if its outgoing
trade cost ( d ln τouti ) falls relative to the incoming trade cost of its export markets, weighted by the importance of
each market for country i’s income (T d ln τ in). In this equation, productivity shocks are pre-multiplied by the ma-
trix M. In contrast, incoming trade cost shocks are pre-multiplied by the matrix T, because they already include the
expenditure share weights (sni), and outgoing trade cost shocks already incorporate the income share weights (tin).
From equation (32), incoming trade cost shocks ( d ln τ in) also directly a�ect welfare through a higher cost of imports,
which raises the cost of living. In addition to these direct e�ects, trade cost shocks like productivity shocks also have
indirect general equilibrium e�ects, through the resulting endogenous changes in incomes.
4.2 Trade Imbalance
Our bilateral friends-and-enemies exposure measures in equations (15) and (19) are derived under the ACR macro
restrictions, including balanced trade. We now show that Propositions 1 and 2 naturally generalize to the case of
exogenous trade imbalances commonly considered in the quantitative international trade literature. We measure the
�ow welfare of the representative agent as per capita expenditure de�ated by the consumption price index:
un =wn`n + dn
`n
[∑Ni=1 p
−θni
]− 1θ
(33)
where dn is the nominal trade de�cit. Market clearing requires that income in each location equals expenditure on
goods produced in that location:
wi`i =N∑n=1
sni[wn`n + dn
]. (34)
Trade Matrices We begin by establishing some properties our trade matrices under trade imbalance. We continue
to use qi ≡ wi`i/
(∑n wn`n) to denote country i’s share of world income. Let ei ≡
(wi`i + dn
) /(∑n wn`n) denote
country i’s share of world expenditures, where we use the fact that the aggregate de�cit for the world as a whole is
equal to zero. Let di ≡ qi/ei denote country i’s income-to-expenditure ratio, which is equal to one divided by one plus
its nominal trade de�cit relative to income. Let D ≡ Diag (d) be the diagonalization of the vector d; note q′ = e′D.
Under trade balance, qi = ei for all i, and D = I.
We continue to use S to denote the expenditure share matrix and T to denote the income share matrix: snicaptures the expenditure share of importer n on exporter i and tin captures the share of exporter i’s income derived
from selling to importer n. Under trade balance, qitin = qnsni, but this is no longer the case under trade imbalance.
Instead, we have the following results.
Lemma 3. Under trade imbalance, q′ = e′S, e′ = q′T. Moreover,
19
1. q′ is the unique left-eigenvector of D−1S with all positive entries summing to one; the corresponding eigenvalue is
one. q′ is also the unique left-eigenvector of TD and TS with eigenvalue equal to one.
2. e′ is the unique left-eigenvector of SD−1 with all positive entries summing to one; the corresponding eigenvalue is
one. e′ is also the unique left-eigenvector of DT and ST with eigenvalue equal to one.
Proof. See Section B.1 of the online appendix.
Comparative Statics Using these properties of the trade matrices, we now derive countries’ income and welfare
exposure to productivity shocks under trade imbalance. As the model does not generate predictions for how trade
imbalances respond to shocks, we follow the common approach in the quantitative international trade literature of
treating them as exogenous. In particular, we assume that trade imbalances are constant as a share of world GDP,
which given our choice of world GDP as the numeraire, corresponds to holding the nominal trade de�cits dn �xed
for all countries n.
Totally di�erentiating (33) and (34), we obtain the following generalizations of equations (13) and (14) to incorpo-
rate trade imbalances:
d lnwi =N∑n=1
tni
(d ln en + θ
(N∑h=1
snh d ln pnh − d ln pni
)), (35)
d lnun = d ln en −N∑m=1
snm d ln pnm. (36)
The introduction of trade imbalance has three main implications for these comparative static relationships. First,
trade imbalances complicate the relationship between the expenditure share (S), income share (T) and cross-substitution
(M) matrices, because with income no longer equal to expenditure for each country (ei 6= qi), we have qitin 6= qnsni.
Second, the market-size e�ect in the income equation depends on changes in expenditure rather than changes in
income (the �rst term in equation (35)). Third, the income e�ect in the welfare equation also depends on changes
in expenditure rather than changes in income (the �rst term in equation (36)). Under our assumption that trade
imbalances stay constant as a share of world GDP, we have the following generalization of our earlier results.
Proposition 9. Assume constant trade de�cits dn for all countries n. The general equilibrium impact of global produc-
tivity shocks on countries’ income and welfare has the following bilateral “friends” and “enemies” matrix representations:
d ln w = W d ln z, W ≡ − θ
θ + 1
(I− TD + θTS
θ + 1+ Q
)−1
M, (37)
d ln u = U d ln z, U ≡ (D− S) W + S, (38)
where recall that D is the diagonalization of the vector of the ratio of income-to-expenditure di.
Proof. The Proposition follows from equations (35) and (36), noting that for all n, d ln dn = 0 =⇒ d ln en =
qnen
d lnwn, as shown in Section F.2 of the online appendix.
4.3 Deviations from Constant Elasticity Import Demand
Our friends-and-enemies measures of countries’ income and welfare exposure to small productivity shocks in equa-
tions (15) and (19) are only exact under the assumption of a constant elasticity import demand system. Using our
20
characterization of the general Armington model in Section 2, we now examine the sensitivity of our exposure mea-
sures to deviations from this constant elasticity assumption. We begin by noting that a constant elasticity import
demand system implies that the cross-price elasticities (θnih) in the market clearing condition (8) are:
θnih =
{(snh − 1) θ if i = h
snhθ otherwise. (39)
Without loss of generality, we can represent the cross-price elasticity of any homothetic demand system as:
θnih =
{(snh − 1) θ + onih if i = h
snhθ + onih otherwise,(40)
where onih captures the deviation from the predictions of the constant elasticity speci�cation (39). Noting that ho-
motheticity implies∑Nk=1 snkonkh = 0, we obtain the following generalizations of our bilateral friend-enemy matrix
representations of the income and welfare e�ects of productivity shocks:
d ln w = T d ln w + (θM + O)× ( d ln w − d ln z) , (41)
d ln U = d ln w − S ( d ln w − d ln z) , (42)
where O is a matrix with entries Oin ≡∑Nh=1 tinonih capturing the average across markets n of these deviations
weighted by the share of country i’s income derived from each market, as shown in Section F.3 of the online appendix.
Using homotheticity, we can rewrite O ≡ ε · O as the product between a scalar ε > 0 and a matrix O with an induced
2-norm equal to one (‖O‖ = 1). By construction, ‖O‖ = ε. Using this representation, we can use results from matrix
perturbation to obtain an upper bound on the sensitivity of income exposure to departures from the constant elasticity
model, as a function of the observed trade matrices and the trade elasticity.
Proposition 10. Let d ln w be the solution to the general Armington model in equation (8) and let d ln w be the solution
to the constant elasticity of substitution (CES) Armington model in equation (15). Then
limε→0
‖ d ln w − d ln w‖ε · ‖ d ln w‖
≤ θ
θ + 1‖ (I−V)
−1 ‖‖I− (W + Q)−1‖. (43)
Proof. See Section B.7 of the online appendix.
Given this upper bound on the sensitivity of income exposure from Proposition 10, we can use equation (42) to
compute the corresponding upper bound on the sensitivity of welfare exposure. All terms on the right-hand side of
equation (43) can be computed using the observed trade matrices and the trade elasticity. Therefore, we can can com-
pute these upper bounds for alternative assumed values of the trade elasticity. An immediate corollary of Proposition
10 is that as the departures from the constant elasticity model become small (ε→ 0), income exposure under a variable
trade elasticity converges towards its value in our constant elasticity speci�cation.
Corollary 3. As the deviations from a constant elasticity import demand system become small (lim ε→ 0), income and
welfare exposure in the general Armington model converge to their values in the constant elasticity of substitution (CES)
Armington model.
Proof. This corollary follows immediately from Proposition 10.
21
From Corollary 3, we can interpret the constant elasticity model as a limiting case of the variable elasticity model.
In the neighborhood of this limiting case, our friends-and-exposure income and welfare exposure measures approxi-
mate those for the variable elasticity model. More generally, from Proposition 10, we can provide an upper bound for
sensitivity of income and welfare exposure to departures from the constant elasticity model that be computed using
the observed trade matrices and assumed values for the trade elasticity.
4.4 Multiple Sectors
Our friends-and-enemies su�cient statistics extend naturally to a multi-sector model with a constant trade elasticity.
For continuity of exposition, we focus on a multi-sector version of the constant elasticity Armington model from
Section 3 above, but the same results hold in the multi-sector version of the Eaton and Kortum (2002) model following
Costinot et al. (2012), as shown in Section F.4 of the online appendix. The preferences of the representative consumer
in country n are now de�ned across the consumption of a number of sectors k according to a Cobb-Douglas functional
form:
un =wn∏K
k=1
[∑Ni=1
(pkni)−θ]−αkn/θ ,
K∑k=1
αk = 1, θ = σ − 1, σ > 1. (44)
where σ > 1 is the elasticity of substitution between country varieties and θ = σ − 1 is the trade elasticity.
Using expenditure minimization, the share of country n’s expenditure in industry k on varieties from country i
takes the standard constant elasticity form:
skni ≡(pkni)−θ∑N
j=1
(pknj)−θ , (45)
and we let tkin ≡ skniαkn wn`nwi`ibe the fraction of exporter i’s income derived from selling to importer n in industry k.
Using the market clearing condition that country income equals expenditure on goods produced by that country,
the impact of small changes in country productivity that are common across industries ( d ln zk` = d ln z` for all k)
on income and welfare in all countries has the following “friends” and “enemies” matrix representation:
d ln w︸ ︷︷ ︸income e�ect
= T d ln w︸ ︷︷ ︸market-size e�ect
+ θM ( d ln w − d ln z)︸ ︷︷ ︸cross-substitution e�ect
, (46)
d ln u︸ ︷︷ ︸welfare e�ect
= d ln w︸ ︷︷ ︸income e�ect
− S ( d ln w − d ln z)︸ ︷︷ ︸cost-of-living e�ect
, (47)
where the expenditure share matrix (S), income share matrix (T) and cross-substitution matrix (M) are now:
Sni ≡K∑k=1
αknskni, Tin ≡
K∑k=1
tkni =K∑k=1
αknskniwn`nwi`i
, Min ≡N∑h=1
K∑k=1
tkihskhn − 1n=i. (48)
As in the single-sector model, Sni equals the aggregate share of importer n’s expenditure on goods produced by
exporter i; Tin is again the aggregate share of exporter i’s income derived from importer n; and Min again captures
the overall competitive exposure of country i to country n through each of their common markets (countries h and
industries k), weighted by the importance of each market for i’s income (tkih).
Our income and welfare exposure measures in the multi-sector model again can be decomposed into the contribu-
tion of di�erent economic mechanisms. From equation (46), productivity shocks a�ect income through the market-size
e�ect, which is captured by the income share matrix T, and the cross-substitution e�ect, which is captured by the
22
matrix M. Similarly, from equation (47), productivity shocks a�ect welfare through the income e�ect and the cost-
of-living e�ect, where this cost-of-living e�ect depends on the expenditure share matrix S. Both the income and
welfare e�ect retain the decomposition into partial and general equilibrium e�ects using the series representation of
the matrix inversion, as in equation (22) for the single-sector model above.
In this multi-sector model, changes in comparative advantage across industries provide an additional source of
terms of trade e�ects between countries. Even common changes in productivity across all sectors have heterogeneous
bilateral e�ects on income and welfare depending on the extent to which pairs of countries share similar patterns of
comparative advantage across industries. Furthermore, we can examine the heterogeneous e�ects of these common
changes in productivity across industries in trade partners using analogous sector-level measure of value-added ex-
posure to global productivity shocks:
d ln Yk = Wk d ln z, (49)
Wk ≡ TkW + θMk (W − I) , (50)
Tkin ≡ tkni, Mk
in ≡N∑h=1
tkihskhn − 1n=i,
where Yk is the vector of value-added in sector k across countries. Aggregating across sectors, our overall income
exposure measure (W) is the weighted average of these sector-level value-added exposure measures (Wk), with
weights equal to sector value-added shares:
Wi =∑k
rki Wki , rki ≡
wi`ki∑K
h=1 wi`hi
, (51)
where Wi is the income exposure vector for country i with respect to productivity shocks in its trade partners n and
Wki is the analogous sector value-added exposure vector for country i and sector k.
4.5 Multiple Sectors and Input-Output Linkages
We now show that we can further generalize this speci�cation with multiple sectors from the previous subsection
to incorporate input-output linkages, following Caliendo and Parro (2015). Again for continuity of exposition, we
focus on a multi-sector version of the constant elasticity Armington model, but the same results hold in a multi-sector
version of the Eaton and Kortum (2002) model, as in Caliendo and Parro (2015).
The representative consumer’s preferences are again de�ned across the consumption of a number of sectors,
as in equation (44) in the previous subsection. Within each sector, each country’s good is produced with labor and
composite intermediate inputs according to a constant returns to scale production technology. These goods are subject
to iceberg trade costs, such that τkni ≥ 1 units must be shipped from country i to country n in sector k in order for one
unit to arrive (where τkni > 1 for n 6= i and τknn = 1). Therefore, the cost to a consumer in country n of purchasing a
good from country i within sector k is:
pkni = τknicki , cki =
(wizki
)γki K∏j=1
(P ji
)γk,ji,
K∑k=1
γk,ji = 1− γki , (52)
where cki denotes the unit cost function for sector k and country i; γki is the share of labor in production costs in sector
k in country i; γk,ji is the share of materials from sector j used in sector k in country i; and zki captures value-added
productivity in sector k in country i.
23
Country income and welfare exposure to global productivity shocks continue to have the “friends” and “enemies”
matrix representation in equations (15) and (19). These exposure measures are again summarized by the expenditure
share (S), income share (T) and cross-substitution (M) matrices. As before, Sni is the expenditure share of consumers
in market n on the value-added of country i, Tin is the share of value-added that country i derives from country n,
and Min is the competitive exposure of country i to country n. However, the elements of these matrices now di�er,
because they use the observed input-output matrix to take into account the full structure of the network.
We now show how the elements of these matrices capture the network structure, with the full derivations reported
in Section F.6 of the online appendix. We use i, n, h, o to index countries and j, k to index industries. The elements of
the expenditure share matrix Sni are now network adjusted as follows:
Sni ≡N∑h=1
K∑k=1
αknsknhΛkhi, (53)
where the �rst summation is across countries h and the second summation is across industries k; αkn is market n’s
Cobb-Douglas expenditure share for industry k; sknh is the share of market n’s expenditure within that industry on
country h; Λkhi captures the share of revenue in industry k in country h that is spent on value-added in country i.
Similarly, the elements of the income share matrix Tin are now also network adjusted as follows:
Tin ≡N∑h=1
K∑k=1
Πkihϑ
khn, (54)
where the �rst summation is across countries h and the second summation is across industries k; Πkih is the network-
adjusted income share that country i derives from selling to industry k in country h; and ϑkhn is the share of revenue
that industry k in country h derives from selling to country n. Finally, the elements of the cross-substitution matrix
are also network adjusted as follows:
Min ≡N∑h=1
K∑k=1
N∑o=1
Πkio
ϑkoh +
N∑j=1
Θkjoh
Υkhon, (55)
where the �rst summation is across countries h, the second summation is across industries k, and the third summation
is across countries o; Πkio is the network-adjusted share of income in country i derived from selling to country o in
industry k; ϑkoh is the share of revenue in industry k in country o that is derived from selling to country h; Θkjoh
captures the fraction of revenue in industry k in country o derived from selling to producers in industry j in country
h; Υknoh captures the responsiveness of country h’s expenditure on industry k in country o with respect to a shock to
costs in country n.
4.6 Economic Geography
Finally, we show that our constant elasticity Armington trade model in Section 3 can be generalized to incorporate
labor mobility across locations, as in models of economic geography, including Helpman (1998), Redding and Sturm
(2008), Allen and Arkolakis (2014), Ramondo et al. (2016) and Redding (2016). The economy as a whole is endowed
with an exogenous measure of workers ¯, each of whom has one unit of labor that is supplied inelastically. Workers
are perfectly mobile across locations, but have idiosyncratic preferences for each location, which are drawn from an
extreme value distribution.
24
As in the Armington trade model without labor mobility, we can use the market clearing condition that equates
income in each location to expenditure on goods produced in that location to examine the impact of productivity
shocks on income in all locations. Unlike the Armington trade model, population in each location in this market
clearing condition is now endogenously determined by a population mobility condition. Using these market clearing
and population mobility conditions, the impact of small productivity shocks on income in all locations again has a
bilateral “friends” and “enemies” matrix representation:
d ln w = T d ln w +
[((σ − 1)− κ
1 + κ
)TS−
(σ − 1
1 + κ
)I +
κ
1 + κS
]( d ln w − d ln z) , (56)
as shown in Section F.7 of the online appendix. Having solved for this impact of the productivity shock on wages
from equation (56), we can use these solutions in the population mobility condition to determine its impact of the
population share of each location (ξn):
d ln ξ = κ (I−Ξ) [ d ln w − S ( d ln w − d ln z)] , (57)
where Ξ is a matrix in which each row is equal to vector of population shares across locations. Population mobility
ensures that the impact of the productivity shock on expected utility (including the idiosyncratic preference shock) is
equalized across all locations. Using our solutions for wages from equation (56) in the population mobility condition,
we also can recover this impact on the common level of expected utility:
d ln u = ξ′ [ d ln w − S ( d ln w − d ln z)] , (58)
where ξ is the vector of population shares of locations.
As in the trade model without labor mobility, income and welfare exposure to productivity shocks depend on
the expenditure share matrix (S), the income share matrix (T) and the product of these two matrices that captures
cross-substitution (TS). In addition, in the economic geography model with labor mobility, both welfare exposure
and the population response to these productivity shocks depend on population shares (though ξ and Ξ).
5 Economic Friends and Enemies
In this section, we report our main empirical results for country income and welfare exposure to productivity shocks.
In Subsection 5.1, we introduce our international trade data. In Subsection 5.2, we examine the quality of the approx-
imation of our linearization to the full non-linear solution of the model for the empirical distribution of productivity
shocks implied by the observed data. In Subsection 5.3, we use our baseline constant elasticity Armington model from
Section 3 to examine global income and welfare exposure to productivity shocks for more than 140 countries over
more than 40 years from 1970-2012. In Subsection 5.4, we compare the predictions of our baseline constant elasticity
Armington model to those of our extensions to incorporate multiple sectors and input-output linkages.
5.1 Data
Our data on international trade are from the NBER World Trade Database, which reports values of bilateral trade be-
tween countries for around 1,500 4-digit Standard International Trade Classi�cation (SITC) codes, as discussed further
in Section H of the online appendix. The ultimate source for these data is the United Nations COMTRADE database
25
and we use an updated version of the dataset from Feenstra et al. (2005) for the time period 1970-2012.5 We augment
these trade data with information on countries’ gross domestic product (GDP), population and bilateral distances
from the GEODIST and GRAVITY datasets from CEPII.6 In speci�cations incorporating input-output linkages, we use
a common input-output matrix for all countries from Caliendo and Parro (2015). We use these datasets to construct the
T, M and S matrices for our three speci�cations of the single-sector constant elasticity Armington model (Section 3),
our multi-sector extension (Section 4.4) and our input-output extension (Section 4.5). In our single-sector model, our
baseline sample consists of an balanced panel of 143 countries over the 43 years from 1970-2012. In our multi-sector
models, we report results aggregating the products in the NBER World Trade Database to 20 International Standard
Industrial Classi�cation (ISIC) industries for which we have input-output data.
5.2 Quality of the Approximation and Computational Burden
We begin by comparing the predictions from our friend-enemy (�rst-order) linearization with those from the conven-
tional exact-hat algebra approach. First, we undertake this comparison using the empirical distribution of productivity
shocks implied by the observed trade data. Second, we report the results from broader comparisons using simulated
productivity shocks. Third, we compare the computational performance of the two approaches.
EmpiricalDistribution of Productivity Shocks To compare our linearization with exact-hat algebra for empirically-
reasonable productivity shocks, we begin by recovering the empirical distribution of productivity and trade cost
shocks that rationalize the observed trade data in our baseline single-sector constant elasticity Armington model.
Note that changes in productivity and trade costs are only separately identi�ed up to a normalization or choice of
units, because an increase in a country’s productivity is isomorphic to a reduction in its trade costs with all partners
(including itself). To separate these two variables, we use the normalization that there are no changes in own trade
costs over time (τnn = 1), which absorbs common unobserved changes in trade costs across all partners into changes
in productivity. But our �ndings for the quality of our approximation are not sensitive to the way in which we recover
productivity shocks, as explored in the Monte Carlo simulations below.
We use this normalization and an assumed standard value of the trade elasticity of θ = 5 to recover changes in trade
costs and productivities (τni, zi) from the model’s gravity equation for bilateral trade �ows and its market clearing
condition that equates a country’s income with expenditure on the goods produced by that country, as discussed
further in Section G.1.1 of the online appendix. In Figure 1a, we display the empirical distribution of log changes in
productivities (ln zi) implied by the observed data from 2000-2010. As apparent from the �gure, we �nd that these
log changes in productivities are clustered relatively closely around their mean of zero, although some individual
countries can experience large changes in log productivities, in part because any common trade cost shocks across all
partners are absorbed into these changes in log productivities.
Having recovered these changes in productivities (zi) implied by the observed trade data, we now compare the
predictions from our (�rst-order) linearization for the impact of these productivity shocks on income to those from the
non-linear exact-hat algebra approach in equation (26). In particular, we set countries’ productivity shocks equal to
their empirical values (zi), undertake an exact-hat algebra counterfactual holding trade costs constant (τni = 1), and5See https://cid.econ.ucdavis.edu/wix.html.6See http://www.cepii.fr/cepii/en/bdd_modele/bdd.asp.
26
solve for the counterfactual changes in countries’ per capita incomes (wi). We compare the results from these exact-
hat algebra counterfactuals to the predictions of our linearization, which implies a log change in countries’ per capita
incomes in response to these productivity shocks of ln w = W ln z. We also undertake an analogous exercise for
changes in bilateral trade costs (τ−θni ), in which we undertake counterfactuals holding productivities constant (zi = 1),
and compare the counterfactual changes in countries’ per capita incomes from the exact-hat algebra counterfactuals
(wi) to the predictions of our linearization, as discussed in Section G.1.1 of the online appendix.
Figure 1: Empirical Distribution of Productivity Shocks and Counterfactual Income Predictions
(a) Distribution Across Countries of Log ProductivityShocks (ln zit) from 2000-2010
-3 -2 -1 0 1 2 3
Inverted Log-Changes in Productivity: 2000-2010
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) Our Approximation Versus Exact-hat CounterfactualPredicted Changes in Income (ln wit) from 2000-2010
-3 -2 -1 0 1 2 3 4
Hat-Algebra
-3
-2
-1
0
1
2
3
4
App
roxi
mat
ion
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
In Figure 1b, we display the predicted log changes in per capita incomes as a result of productivity shocks from
our linearization and the exact-hat algebra counterfactuals. Although the two sets of predictions are not exactly
the same as one another, we �nd an extremely strong relationship between them, such that they are almost visibly
indistinguishable, with a correlation coe�cient of more than 0.999. From Section 3.6 above, our approximation is exact
under autarky (snn → 1 and tii → 1) and under free trade, and performs well using matrices of random trade shares
scaled to sum to one. Empirically, we �nd that observed trade matrices are well approximated by a weighted average
of autarky, free trade and random matrices, and hence our approximation also performs well using observed trade
matrices. In Section G.1.1 of the online appendix, we report an analogous comparison for bilateral trade cost shocks.
Although we again �nd a strong relationship between the predictions of our linearization and the exact-hat algebra
counterfactuals, it is weaker than for productivity shocks. An important reason for this di�erence is that productivity
shocks are common across all trade partners, which means that the direct e�ect of these productivity shocks can be
taken outside of the summation across trade partners into a separate �rst term that is identical in our linearization
and the exact hat algebra in equations (26) and (27). In contrast, the direct e�ect of bilateral changes in trade costs
cannot be taken outside of this summation sign, because it varies across trade partners.
Simulated Productivity Shocks To explore the robustness of these results, we next report a broader set of compar-
isons between our linearization and the full non-linear solution of the model using simulated productivity shocks. In
particular, we undertake 1,000 Monte Carlo simulations in which we draw (with replacement) productivity shocks for
each country from the empirical distribution of productivity shocks from 2000-2010. Using these simulated produc-
tivity shocks, we undertake exact-hat algebra counterfactuals to compute predicted log changes in per capita income,
27
and compare these predictions with those from our linearization. In Figure 3, we show the distribution of regression
slope coe�cients and R-squared between the two sets of predictions. Across all of our simulations, we �nd slope
coe�cients from 0.99-1.01 and correlation coe�cients of more than 0.999.
As a further robustness check, we multiplied the size of the productivity shocks by 1,000, and undertook another
1,000 Monte Carlo simulations. Even with productivity shocks three orders of magnitude larger than those implied
by the observed trade data, we continue to �nd the same pattern of results, with a correlation coe�cient of above 0.99
in all of our simulations. To explore the sensitivity of these results with respect to our assumed trade elasticity, we
experimented with values for the trade elasticity from 2 to 20, which spans the empirically-relevant range of values
for this parameter. Even for trade elasticities as extreme as 2 and 20, we continue to �nd regression slope coe�cients
ranging from 0.85-1.10 and correlation coe�cients of above 0.99, as reported in Section G.1.1 of the online appendix.
Taken together, these results suggest that our friend-enemy income exposure measures for productivity shocks are
close to exact for empirically-reasonable changes in productivities and values of trade elasticities.
Bounds on Approximation Error Further light on these empirical results comes from our analytical results for
the quality of the approximation in Propositions 3-7 in Subsection 3.6 above. In Table G.1 in Section G.1.1 of the
Online Appendix, we report the distribution of Hessian eigenvalues over our sample period. We �nd that even the
largest eigenvalue of the Hessian matrix (Hfi ) is close to zero for each country. Therefore, as we approximate the
log-income change for each country i separately, the second-order term εi, when maximized by a country-speci�c
vector of TFP shocks{zmax,i
}, accounts for at most a tiny fraction of the variation in ln wi. For example, for the year
2000 and on average over time, we �nd that the second-order approximation error for the income exposure of each
country is bounded by 0.26 percent and 0.36 percent of the variance of productivity shocks respectively.
Furthermore, for all countries, we �nd that the second-largest eigenvalues µ2ndi are substantially closer to zero,
which implies that any productivity shock vector that is orthogonal to zmax,i generates approximately zero second-
order e�ects. We further �nd that the standard unit vector e` comes close to achieving the upper bound for the `-th
equation, i.e. e` ≈ zmax,` for all `. Hence, the second-order term for evaluating the e�ect of a productivity shock in
country ` on income in country i 6= ` is small (approximately bounded by∣∣µ2nd,i
∣∣) even relative to the own-e�ect on
country ` itself, which is already small (approximately bounded by∣∣µmax,i
∣∣). Even considering all higher-order terms
together (second-order and above) in Proposition 7, and using the assumption that the Hessian eigenvalues evaluated
over the support of the distribution of productivity shocks are bounded by the Hessian eigenvalues observed during
our sample period, we continue to �nd that the approximation error remains small. In particular, we �nd that the global
approximation errors for income exposure to own productivity shocks are less than 0.62 percent of the variance of
productivity shocks, and that these global approximation errors for income exposure to other countries’ productivity
shocks are 0.33 percent of the variance of productivity shocks.
Computational Speed In comparing our (�rst-order) linearization to the exact-hat algebra, another relevant cri-
terion alongside the quality of the approximation is the relative computational burden. We compare computational
speed for the two approaches using Matlab, a single thread (virtual CPU core), and a tolerance of 10−6 for solving
the full non-linear solution of the model using the exact-hat algebra (our matrix inversion uses machine precision).
We compute 6,149 comparative statics for country productivity shocks (143 countries × 43 years) for our baseline
28
Figure 2: Distributions of Regression Slope Coe�cients and Coe�cients of Correlation Comparing our Friend-EnemyApproximation to Exact-Hat Algebra Predictions in Monte Carlos using Simulated Productivity Shocks (Trade Elas-ticity θ = 5)
0.995 1 1.005 1.01Regression Coefficient
0
0.05
0.1
0.15
0.9998 0.99985 0.9999 0.99995 1Coefficient of Correlation
0
0.05
0.1
0.15
Source: NBER World Trade Database and authors’ calculations authors’ calculations using our baseline constant elasticity Armington model fromSection 3. Monte Carlo simulations using 1,000 replications. Simulated productivity shocks drawn (with replacement) from the empiricaldistribution of productivity shocks from 2000-10.
single-sector Armington model from Section 3. On both our laptops and high-performance computer servers, we
�nd that our linearization is around 70,000 times faster than the exact-hat algebra.7 As we move from our baseline
single-sector speci�cation to the more computationally demanding input-output speci�cation, we �nd that this dif-
ference in processing time increases further. As a result of these improvements in computational speed, it becomes
feasible to compare the results of large numbers of counterfactuals across di�erent quantitative models, such as our
single-sector, multi-sector and input-output speci�cations. Therefore, in settings in which large numbers of coun-
terfactuals must be undertaken, our linearization can provide a useful complement to solving for the full non-linear
solution using exact-hat algebra. At the very least, using the predictions of our linearization as the initial guess for
the full model solution brings dramatic improvements in computational speed. More broadly, our approach closely
approximates the full model solution, has a clear interpretation in terms of the underlying economic mechanisms, and
enables researchers to easily explore the sensitivity of counterfactual predictions across di�erent quantitative models.
5.3 Aggregate Income and Welfare Exposure 1970-2012
We now present our main empirical results on global income and welfare exposure to productivity shocks using our
baseline constant elasticity Armington model from Section 3. First, we present results for the overall distribution of
income and welfare exposure across countries and over time. Second, we provide further evidence on the large-scale
changes in bilateral networks of income and welfare exposure that have occurred over our sample period. Third,
we evaluate the role of general equilibrium relative to partial equilibrium e�ects in shaping the impact of these pro-
ductivity shocks. Fourth, we examine the di�erent economic mechanisms of the market-size, cross-substitution and
cost-of-living e�ects. Fifth, we investigate the contributions of importer, exporter and third market e�ects in shaping
countries’ exposure to foreign productivity shocks.7To solve the exact-hat algebra counterfactuals, we use an iterative algorithm of the form considered in Allen and Arkolakis (2014) and Allen
et al. (2020). Using a standard Matlab optimization routine such as Fsolve substantially increases the computation time for these counterfactuals.Allowing for parallelization (multiple virtual CPU cores) reduces this computation time.
29
5.3.1 Global Income and Welfare Exposure
Using our baseline constant elasticity Armington model, we compute bilateral income and welfare exposure to pro-
ductivity shocks for our balanced panel of 143 countries over the 43 years from 1970-2012 (143×143×43 = 879, 307
bilateral predictions for each variable). In Figure 3, we show mean income and welfare exposure to foreign produc-
tivity shocks over time (excluding own productivity shocks) and their 95 percent con�dence intervals. In interpreting
the magnitudes, note that these values correspond to mean changes in income and welfare with respect to in�nites-
imally small productivity shocks. Once we take into account the empirical size of productivity shocks, we obtain
predicted changes in income and welfare comparable to those from the full non-linear model solution, as shown in
Section 5.2 above. Given our choice of world GDP as numeraire, a productivity shock that raises a country’s own per
capita income tends to reduce the per capita income of other countries (in order to hold world GDP constant), which
results in a negative average income exposure (left panel). As our choice of numeraire holds world GDP constant
over time, we also �nd that mean income exposure is relatively �at over time. Besides raising a country’s own per
capita income, a productivity shock also reduces its prices, and we �nd that this cost of living e�ect is su�ciently
strong that average welfare exposure is positive (right panel). These welfare results are invariant to the choice of
numeraire, which cancels from the income and cost of living components of welfare, as discussed above. One striking
feature of the �gure is the substantial and statistically signi�cant increase in average welfare exposure over time, by
around 72 percent from 1970-2012. This pattern of results is consistent with the view that the increased globalization
that occurred over our sample period enhanced countries’ interdependence on one another, as captured by average
exposure to foreign productivity growth.
Another striking feature of Figure 3 is the substantial dispersion in exposure to foreign productivity shocks, as
re�ected in the 95 percent con�dence intervals. In Figure 4, we provide further evidence on this heterogeneity in
welfare exposure using Box and Whisker plots, in which the interquartile range is shown by the edges of the box,
and the extended lines represent the 5th and 95th percentiles. Although on average foreign productivity shocks raise
importer welfare, an importer at the 5th percentile experiences a reduction in welfare only somewhat smaller than the
increase in welfare enjoyed by an importer at the 95th percentile. Furthermore, we �nd an increase in the dispersion
of welfare exposure from the early 1980s onwards in Figure 4, which suggests that increased globalization has not
only raised countries average exposure to foreign productivity growth, but also enhanced the inequality in the e�ects
of this productivity growth, namely the extent to which individual countries are winners or losers from expansions
in the productive capacity of particular trade partners.8
In our constant elasticity Armington model, this heterogeneity in welfare exposure re�ects the interaction of the
cross-substitution, market-size and cost-of-living e�ects. First, the direct e�ect of a country’s productivity growth in
lowering its prices has a negative cross-substitution e�ect on the per capita income of its trade partners, as these trade
partners face increased competition in markets around the world. Second, this direct e�ect of productivity growth in
lowering prices also raises welfare in all countries through a lower cost of living. Third, productivity growth raises a
country’s own per capita income, which has a positive market-size e�ect on the per capita income of other countries.
Fourth, the resulting endogenous changes in per capita income in all countries have further indirect e�ects on prices,
income and welfare through these cross-substitution, market-size and cost-of-living mechanisms. The relative balance8The step increase in the dispersion of welfare exposure between 1999 and 2000 in Figure 4 is driven by a step increase in the number of bilateral
importer-exporter pairs with positive international trade �ows between those years.
30
Figure 3: Mean Income and Welfare Exposure to Productivity Shocks in Other Countries over Time
-.006
8-.0
066
-.006
4-.0
062
-.006
-.005
8M
ean
Inco
me
Expo
sure
1970 1980 1990 2000 2010Year
Mean 95 percent confidence interval
.000
05.0
001
.000
15.0
002
Mea
n W
elfa
re E
xpos
ure
1970 1980 1990 2000 2010Year
Mean 95 percent confidence interval
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
of all of these forces depends on the geography of trade �ows, as re�ected in the expenditure share matrix (S), the
income share matrix (T), and the cross-substitution matrix (M).
In Figure 5, we illustrate global welfare exposure to productivity shocks in 1970, 1985, 2000 and 2012 using a
network graph, where the nodes are countries and the edges capture bilateral welfare exposure. For legibility, we
display the 50 largest countries in terms of GDP and the 200 edges with the largest absolute values of bilateral welfare
exposure.9 The size of each node captures the importance of each country as a source of productivity shocks (as
a source of welfare exposure for other countries); the arrow for each edge shows the direction of bilateral welfare
exposure (from the source of the productivity shock to the exposed country); and the thickness of each edge shows
the absolute magnitude of the bilateral welfare exposure. Countries are grouped to maximize modularity (the fraction
of edges within the groups minus the expected fraction if the edges were distributed at random).
At the beginning of our sample period in 1970, the global network of welfare exposure is dominated by the U.S.,
Germany and other Western industrialized countries (top-left panel). Moving forward to 1985, we see the emergence
of Japan and a cluster of Newly Industrialized Countries (NICs) in Asia, and we observe Western Europe increasingly
emerging as a separate cluster of interdependent nations. By the time we reach 2000, the separate clusters of countries
in Asia and Western Europe become even more apparent, with China beginning to displace Japan at the center of the
Asian cluster. By the end of our sample period in 2012, China replaces the U.S. at the center of the global network
of welfare exposure, with the US more tightly connected to China and other Asian countries than to the cluster of
Western European countries. Therefore, we �nd substantial changes, not only in the mean and dispersion of welfare
exposure, but also in the network of bilateral interdependencies between countries.9All of the bilateral welfare exposure links shown in the �gure are positive.
31
Figure 4: Box-Whisker Plot of Distribution of Welfare Exposure over Time
-.000
050
.000
05.0
001
Wel
fare
Exp
osur
e
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3; box showsinterquartile range; extended lines show 5th and 95th percentiles.
Figure 5: Global Welfare Exposure, 1970, 1985, 2000 and 2012
(a) 1970 (b) 1985
(c) 2000 (d) 2012
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
32
5.3.2 Regional Networks of Welfare Exposure
To provide further evidence on changes in regional networks of welfare exposure over our sample period, we use
chord or radial network diagrams, as used for example in comparative genomics in Krzywinski et al. (2009) and for
bilateral migration �ows in Sander et al. (2014).
In Figure 6, we show welfare exposure in 1970 and 2012 for U.S., Canada, Mexico, Japan and China, where each
country is labelled by its three-letter International Organization for Standardization (ISO) code.10 These countries are
arranged around a circle, where the size of the inner segment for each country shows its overall outward exposure (the
e�ect of its productivity shocks on other countries), and the gap between the inner and outer segments shows its over-
all inward exposure (the e�ect of foreign productivity shocks upon it). Arrows emerging from the inner segment for
each country show the bilateral impact of its productivity shocks on welfare in other countries. Arrows pointing to-
wards the gap between the inner and outer segments show the bilateral impact of other countries’ productivity growth
on its welfare.11 In 1970, the network is dominated by the e�ect of US productivity shocks on welfare in the other
countries, and Japan is substantially more connected to the network than China. By 2012, following Mexican trade
liberalization in 1987, the Canada-US Free Trade Agreement (CUSFTA) in 1988 and the North American Free Trade
Agreement (NAFTA) in 1994, we observe much deeper integration between the three North American economies.
Additionally, we �nd a reversal of the relative positions of the two Asian economies, with China substantially more
integrated into the network than Japan.
Figure 6: North American Welfare Exposure, 1970 and 2012
(a) 1970 (b) 2012
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
In Figure 7, we display welfare exposure for a broader group of Asian countries. Three features stand out. First,
we again �nd a dramatic change in the relative positions of Japan and China. Whereas in 1970 Japan dominated10To ensure a consistent treatment of countries over time, we manually assign some three-letter codes, such as the code USR for the members
of the former Soviet Union.11We omit own exposure to focus on the impact of foreign productivity shocks on country welfare. Almost all values of our welfare exposure
measure in these diagrams are positive. For ease of interpretation, we add a constant to our welfare exposure measure in each year, such that itsminimum value is zero, which implies that these diagrams show the impact of the productivity shock on relative levels of welfare.
33
the network of welfare exposure, in 2012 this position is �rmly occupied by China. Second, Vietnam becomes both
substantially more exposed to foreign productivity shocks and a much more important source of these productivity
shocks for other countries, following its trade liberalization. Third, the overall network of welfare exposure is much
denser in 2012 than in 1970, consistent with greater trade integration among these Asian countries increasing their
economic interdependence on one another. In Section G.1.2 of the online appendix, we provide further evidence on
large-scale changes in regional networks of welfare exposure for Central Europe following the fall of the Iron Curtain.
Figure 7: Asian Welfare Exposure, 1970 and 2012
(a) 1970 (b) 2012
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
5.3.3 Partial and General Equilibrium E�ects
We now examine the role of partial and general equilibrium forces in the model using our series decomposition in
equation (22) above. From our earlier discussion, the direct or partial equilibrium e�ect of productivity growth in a
given exporter is to increase its price competitiveness in all markets, which leads to substitution away from all other
countries’ goods. But there are also indirect or general equilibrium e�ects, as the endogenous changes in per capita
income that occur in response to this productivity growth also a�ect both cross-substitution and market demand.
In Figures 8a and 8b, we show this series decomposition for the impact of Chinese productivity growth on U.S.
income and welfare respectively. In both �gures, the thick blue line shows the partial equilibrium e�ect (the �rst-
order term θθ+1M in the series-decomposition in equation (22)). The thinner blue line immediately below adds to
this partial equilibrium e�ect the �rst term from the general equilibrium component of the series decomposition (the
term θθ+1MV in equation (22)). Each of the additional thinner blue lines further below add successively higher-order
terms from the general equilibrium component of the series decomposition. As we add these additional higher-order
terms, predicted income exposure converges towards our overall income exposure measure in equation (17).
As apparent from the �gure, we �nd that the general equilibrium forces in the model are large relative to the
partial equilibrium forces, and we �nd relatively rapid convergence, such that the addition of a few higher-term terms
in the series decomposition takes us relatively close to our overall measure of income exposure. Taken together, these
results highlight the importance of general equilibrium forces in this class of constant elasticity trade models, and
34
suggest that a misleading picture about the e�ects of productivity growth would be obtained by focusing solely on
the partial equilibrium term of productivity growth weighted by the trade elasticity and the initial expenditure shares.
5.3.4 Cross-Substitution, Market Size and Cost of Living E�ects
We now examine the contributions of the di�erent economic mechanisms in the model by separating out overall
income exposure into the contributions of the market-size and cross-substitution e�ects, and breaking down the
welfare e�ect into the contributions of the income and cost of living e�ects.
Figure 8: Partial and General Equilibrium E�ects of the Impact of Productivity Growth in China (Exporter) on Incomein the United States (Importer) over Time
(a) Income Exposure
1970 1975 1980 1985 1990 1995 2000 2005 2010 2015-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
China-USA W: PE EffectChina-USA W: GE Effect
(b) Welfare Exposure
1970 1975 1980 1985 1990 1995 2000 2005 2010 20150
0.002
0.004
0.006
0.008
0.01
0.012
China-USA U: PE EffectChina-USA U: GE Effect
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
In Figure 9, we illustrate these relationships for a Chinese productivity shock in 2010, where the circles correspond
to each of the other countries in our sample (excluding China). In the top-left panel, we show the relationship between
the cross-substitution e�ect (WSub) and the market size e�ect (W−WSub). We �nd that the cross-substitution e�ect is
always negative, as higher Chinese productivity reduces the competitiveness of other countries in all markets, which
leads consumers in all markets to substitute away from these other countries’ goods, and lowers their per capita
income relative to China. In contrast, the market-size e�ect can be either positive or negative. On the one hand, higher
productivity in China raises its per capita income, which increases the market demand for other countries’ goods, and
increases their income. On the other hand, the reduction in income in other markets from the cross-substitution e�ect
lowers market demand for other countries’ goods, which decreases their income. The overall income e�ect is the net
outcome of these cross-substitution and market-size forces and hence can be either positive or negative. As apparent
from the top-left panel, we �nd a strong relationship between the market-size and cross-substitution e�ects, because
the gravity structure of international trade jointly determines the share of exporter i in importer n’s expenditure (sni)
and the share of importer n in exporter i’s income (tin), which are the key determinants of the relative magnitude of
these two e�ects.
In the top-right panel, we display the cross-substitution e�ect against the overall income e�ect, while in the
bottom-left panel, we show the market-size e�ect against the overall income e�ect. As the cross-substitution e�ect
lowers per capita income, while the market size e�ect increases per capita income, we �nd a negative relationship
in the top-right panel and positive relationship in the bottom left panel. Both the cross-substitution and market-size
35
e�ects are quantitatively relevant relative to the overall income income, with a regression slope coe�cient of -2.87 and
R-squared of 0.46 in the top-right panel, and a regression slope coe�cient of 3.87 and R-squared of 0.61 in the bottom-
left panel. As we consider productivity shocks for di�erent exporters and years, we �nd that exporter country size
plays a central role in driving the relative importance of the market-size and cross-substitution e�ects in the overall
income e�ect. In particular, the market size e�ect is smaller relative to the overall income e�ect for exporters with
smaller shares of world GDP.
In the bottom-right panel, we show the welfare e�ect against the income e�ect, where these two e�ects di�er
from one another through the cost-of-living e�ect. As apparent from this panel, we �nd a positive and statistically
signi�cant relationship between the two variables, with a regression slope coe�cient of 0.24. However, we �nd that
this correlation is far from perfect, with a regression R-squared of only 0.28. This pattern of results highlights the
strength of the cost-of-living e�ect in the model and emphasizes that caution is warranted in making inferences
about changes in welfare from information on changes in income alone.
Figure 9: Mechanisms Underlying the Income and Welfare E�ects
-.05
0.0
5.1
.15
Mar
ket-S
ize
Effe
ct
-.25 -.2 -.15 -.1 -.05Cross-Substitution Effect
Note: Slope coefficient: -1.1615; standard error: 0.0160; R-squared: 0.9781.
Cross-Substitution Versus Market-Size Effects (China Exporter, 2010)-.2
5-.2
-.15
-.1-.0
5C
ross
-Sub
stitu
tion
Effe
ct
-.12 -.11 -.1 -.09 -.08 -.07Income Effect
Note: Slope coefficient: -2.8712; standard error: 0.4996; R-squared: 0.4636.
Cross-Substitution Effect Versus Income Effect (China Exporter, 2010)
-.1-.0
50
.05
.1.1
5M
arke
t-Siz
e Ef
fect
-.12 -.11 -.1 -.09 -.08 -.07Income Effect
Note: Slope coefficient: 3.8712; standard error: 0.4996; R-squared: 0.6111.
Market-Size Effect Versus Income Effect (China Exporter, 2010)
0.0
05.0
1.0
15.0
2W
elfa
re E
ffect
-.12 -.11 -.1 -.09 -.08 -.07Income Effect
Note: Slope coefficient: 0.2390; standard error: 0.0547; R-squared: 0.2782.
Welfare Effect Versus Income Effect (China Exporter, 2010)
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
5.3.5 Own and Third Market E�ects
Even within the cross-substitution e�ect, our approach highlights that higher exporter productivity growth a�ects
importer income and welfare through multiple channels of increased exporter price competitiveness in the importer’s
market, the exporter’s market and third markets.
In Figure 10, we illustrate the contributions of these di�erent terms towards the partial equilibrium cross-substitution
e�ect for US exposure to Chinese productivity growth. Consistent with the emphasis in reduced-form empirical stud-
ies on impacts in the U.S. market, we �nd that much of the direct e�ect of higher Chinese productivity growth occurs
36
within the U.S. (importer’s) market. However, we �nd that a substantial component also occurs within the Chinese
(exporter’s) market, which highlights the role of both U.S. imports and exports in shaping the impact of the China
shock. In comparison, we �nd smaller third market e�ects, with the largest third market e�ects occurring in Singa-
pore, Canada and Japan. This pattern of third market e�ects is intuitive, as this partial equilibrium cross-substitution
e�ect for the U.S. depends on the product of the share of U.S. income derived from a market (tih) and the share of
that market’s expenditure on China (shn). In line with this intuition, Canada is one of the largest markets for the U.S.
(high tih). Although Singapore and Japan are smaller markets for the U.S. (lower tih), they have relative high shares of
expenditure on China (high snh), and hence increased Chinese competitiveness has a large impact on US sales within
these markets.
While, for ease of interpretation, we illustrate the contributions of the importer’s market, exporter’s market and
third markets using the partial equilibrium cross-substitution e�ect, the more subtle general equilibrium interactions
that occur through the cross-substitution and market-size e�ects also take place in these three groups of markets, as
discussed in Section 3.5 and captured in our overall exposure measures above.
Figure 10: USA Exposure to Partial Equilibrium Cross-Substitution E�ect from China, 2010
(a) Importer, Exporter and Third Markets
0 .002 .004 .006Cross-Substitution Effect
Effect in 3rd markets
Effect in exporter market
Effect in own market
(b) Individual Third Markets
0 .00002 .00004 .00006 .00008Cross-Substitution Effect
NLD
GBR
DEU
MEX
TWN
KOR
MYS
JPN
CAN
SGP
Source: NBER World Trade Database and authors’ calculations using our baseline constant elasticity Armington model from Section 3.
5.4 Sectoral Linkages and Income and Welfare Exposure 1970-2012
We now present evidence from our extensions to incorporate multiple sectors and input-output linkages. In Subsection
5.4.1, we compare the aggregate predictions of the single-sector, multi-sector and input-output speci�cations, exam-
ining the extent to which introducing industry comparative advantage and input-output linkages a�ects aggregate
predictions for income and welfare. In Subsection 5.4.2, we examine the new disaggregated sector-level predictions
of these extensions, in which even productivity shocks that are common across sectors in one country can have het-
erogeneous e�ects on sectors in its trade partners, depending on the extent to which they compete with one another
in sector output markets or source intermediate goods from one another in input markets.
37
5.4.1 Aggregate Income and Welfare Exposure
In our baseline single-sector Armington model, changes in country welfare in response to foreign productivity shocks
occur through changes in the factoral terms of trade. In contrast, in our multi-sector extension from Section 4.4, the
terms of trade is also in�uenced by patterns of comparative advantage across sectors. In particular, the e�ect of
productivity growth in any one foreign country on home welfare depends on the extent to which that foreign country
has a similar or dissimilar pattern of comparative advantage across sectors to the home country. In our input-output
extension from Section 4.5, this e�ect of foreign productivity growth on home welfare is further complicated, because
domestic comparative advantage across sectors now depends in part on foreign comparative advantage across sectors,
through the input-output structure of production.
These di�erences in the determinants of the terms of trade in the three models are re�ected in the elements of the
trade matrices (S, T and M) that determine the cross-substitution, market-size and cost-of-living e�ects. Therefore,
although all of these models share the same friends-and-enemies representation of income and welfare exposure, the
way in which this representation is constructed di�ers between them. In the multi-sector model, the elements of the
S and T matrices in equation (48) now depend on the product of the share of country n’s overall expenditure on
industry k (αkn) times its share of expenditure on country i within that industry (skni). If industries di�er in size and
vary in importance in the trade between di�erent bilateral pairs of countries as a result of comparative advantage, the
resulting elements of these matrices di�er from those in the single-sector model. These di�erences in the elements
of the S and T matrices in turn induce corresponding di�erences in the elements of the M matrix, which depend on
the products of sknhtkhi for all markets h. In the input-output model, the elements of all three matrices in equations
(53), (54) and (55), respectively, must be further adjusted to take into account the network structure of production,
using the observed industry-to-industry �ows in the input-output matrix. For the S and T matrices that capture
the share of an importer’s expenditure on each exporter and the share of an exporter’s income derived from each
importer, respectively, this is largely a matter of accounting. We take into account that the gross value of trade from
exporter i to importer n in industry k includes not only the direct value-added created in this exporter and industry
but also indirect value-added created in previous stages of production. For the M matrix, this adjustment also takes
into account that the e�ect of a foreign productivity shock di�ers depending on whether it reduces intermediate input
costs or competitors’ output prices.
We now examine the implications of these di�erences in the S, T and M matrices across the three models for
their aggregate predictions for countries’ welfare exposure to foreign productivity shocks. In our input-output model,
we use a common input-output table for all countries from CP, which implies common industry expenditure shares
across all countries in traded and non-traded sectors. To ensure a fair comparison across the three models, we make
the same assumption of common industry expenditure shares across countries in the multi-sector model, and we
incorporate non-traded goods into all three of our models.12 In Figure 11, we display countries’ welfare exposure to
a Chinese productivity shock in 2010 (excluding China’s own exposure). In the top-left panel, we show the multi-
sector model versus the single-sector model; in the top-right panel, we display the input-output model versus the
single-sector model; and in the bottom-left panel, we report the input-output model versus the multi-sector model.
We �nd strong correlations between the aggregate predictions of all three models, which are statistically signi�cant12Therefore, our single-sector model in this section features a single traded sector and a single non-traded sector, whereas our multi-industry
and input-output models incorporate many disaggregated sectors.
38
at conventional critical values. In the top-left panel, we �nd that aggregate welfare responds more strongly to foreign
productivity growth in the multi-sector model than the single-sector model (slope coe�cient of 1.42), consistent with
an additional margin of adjustment in the multi-sector model (industry comparative advantage). In the top-right
and bottom-left panels, we �nd even stronger responses of aggregate welfare to foreign productivity growth in the
input-output model than in either of the other models (slope coe�cients of 3.18 and 2.23 respectively), consistent with
input-output linkages magnifying the e�ects of productivity improvements.
Figure 11: Aggregate Welfare Exposure in the Single-Sector, Multi-Sector and Input-Output Models
0.0
01.0
02.0
03.0
04W
elfa
re E
xpos
ure,
Mul
ti-se
ctor
0 .001 .002 .003Welfare Exposure, Single-Sector
45 degree line Fitted
Note: Slope coefficient: 1.4176; standard error: 0.0458.
Welfare Exposure, Multi-sector Versus Single-Sector, China Shock 2010
0.0
02.0
04.0
06.0
08.0
1W
elfa
re E
xpos
ure,
Inpu
t-Out
put
0 .001 .002 .003Welfare Exposure, Single-Sector
45 degree line Fitted
Note: Slope coefficient: 3.1807; standard error: 0.1800.
Welfare Exposure, Input-Output Versus Single-sector, China Shock 2010
0.0
02.0
04.0
06.0
08.0
1W
elfa
re E
xpos
ure,
Inpu
t-Out
put
0 .001 .002 .003 .004Welfare Exposure, Multi-Sector
45 degree line Fitted
Note: Slope coefficient: 2.2324; standard error: 0.1393.
Welfare Exposure, Input-Output Versus Multi-sector, China Shock 2010
Source: NBER World Trade Database and authors’ calculations using the single-sector model from Section 3, the multi-sector model from Section4.4, and the input-output model from Section 4.5.
In Figure 12, we examine the impact of Chinese productivity growth on aggregate U.S. income and welfare over
our whole sample period. To ensure that our results for income exposure are invariant to the choice of numeraire,
we display the income e�ect relative to the income-weighted average for all OECD countries. Consistent with our
results for all pairs of countries above, we again �nd similar aggregate welfare predictions across all three models.
In each case, we �nd that Chinese productivity growth has an increasingly negative e�ect on aggregate US income
relative to the OECD average, but an increasingly positive e�ect on aggregate US welfare, highlighting the powerful
cost of living e�ect in these quantitative trade models. Comparing the single-sector and multi-sector models, we �nd
a substantially more negative e�ect of Chinese productivity growth on US relative income once we take industry
specialization into account. As we move from the multi-sector model to the input-output model, we �nd a much
larger positive e�ect of Chinese productivity growth on US welfare, again highlighting the potential for input-output
linkages to magnify the e�ects of productivity improvements.
39
Figure 12: Aggregate Income and Welfare Exposure to Chinese Productivity Growth
-.002
5-.0
02-.0
015
-.001
-.000
50
Inco
me
Effe
ct
1970 1980 1990 2000 2010Year
Income Exposure Relative to OECD Average (USA Importer, China Exporter)
0.0
002
.000
4.0
006
.000
8W
elfa
re E
ffect
1970 1980 1990 2000 2010Year
Welfare Exposure (USA Importer, China Exporter)
Input-Output Multi-Sector Single-Sector
Source: NBER World Trade Database and authors’ calculations using the single-sector model from Section 3, the multi-sector model from Section4.4, and the input-output model from Section 4.5.
5.4.2 Sector Income Exposure
Even though all three models have relatively similar aggregate predictions for income and welfare, the multi-sector
and input-output models have additional disaggregated predictions for sector income, as summarized for the multi-
sector model in equation (50) in Section 4.4 above. In this section, we brie�y illustrate these disaggregated predictions
for sector income by considering the impact of Chinese productivity growth on nearby South-East Asian countries
and other resource-rich emerging economies, using the input-output model from Section 4.5 above. As discussed for
the aggregate income e�ect above, our choice of world GDP as numeraire implies that a productivity shock that raises
a country’s own income tends to reduce income in other countries (in order to hold world GDP constant).
Figure 13: Industry Sales Exposure to Chinese Productivity Growth FOR
CommunicationTextileOffice
ElectricalMedical
Petroleum
-.2-.1
0.1
.2In
dust
ry T
otal
Inco
me
Effe
ct
2000 2005 2010 2015
Korea
Textile
CommunicationMetal products
Electrical
PetroleumMining
-.2-.1
0.1
.2In
dust
ry T
otal
Inco
me
Effe
ct
2000 2005 2010 2015
Singapore
OfficeCommunicationBasic metals
ElectricalMedical
Petroleum
-.20
.2.4
Indu
stry
Tot
al In
com
e Ef
fect
2000 2005 2010 2015
Taiwan
CommunicationTextileTransport (excl. auto)
Electrical
OfficePetroleum
-.2-.1
0.1
.2In
dust
ry T
otal
Inco
me
Effe
ct
2000 2005 2010 2015
Thailand
Source: NBER World Trade Database and authors’ calculations using the input-output model from Section 4.5.
40
For both the nearby South-East Asian countries (Figure 13) and the resource-rich emerging economies (Figure 14)
we �nd some of the most negative e�ects for the Textiles sector. In contrast, we �nd striking di�erences between
the two groups of countries in the sectors with the most positive or least negative income e�ects. For the nearby
South-East Asian countries, the sectors that bene�t most from Chinese productivity growth include the Electrical,
Medical and O�ce sectors, consistent with input-output linkages between related sectors through global value chains
in Factory Asia. However, for the resource-rich emerging economies, the sectors that bene�t most include the Mining,
Agricultural and Basic Metals sectors, consistent with a form of “Dutch Disease,” in which the growth of resource-
intensive sectors propelled by Chinese demand competes away factors of production from less resource-intensive
sectors. Taken together, this pattern of results highlights that even common productivity growth across sectors can
have subtle and heterogeneous e�ects on individual sectors in foreign trade partners, depending on patterns of com-
parative advantage and input sourcing.
Figure 14: USA Industry Sales Exposure to Chinese Productivity Growth
OfficeCommunicationElectrical
Mining
PetroleumAgriculture
-.14
-.12
-.1-.0
8-.0
6-.0
4In
dust
ry T
otal
Inco
me
Effe
ct
2000 2005 2010 2015
Brazil
CommunicationTextileOffice
Basic metals
Mining
Paper
-.2-.1
5-.1
-.05
0In
dust
ry T
otal
Inco
me
Effe
ct
2000 2005 2010 2015
Chile
TextileOfficeCommunication
PetroleumMining
Agriculture
-.15
-.1-.0
50
Indu
stry
Tot
al In
com
e Ef
fect
2000 2005 2010 2015
Former-USSR
Textile
OfficeCommunication
Basic metals
Mining
Food
-.2-.1
5-.1
-.05
0.0
5In
dust
ry T
otal
Inco
me
Effe
ct
2000 2005 2010 2015
South-Africa
Source: NBER World Trade Database and authors’ calculations using the input-output model from Section 4.5.
6 Economic and Political Friends and Enemies
We now use our bilateral measures of exposure to productivity shocks to provide evidence on the political economy
debate about the extent to which increased con�ict of economic interests between countries necessarily involves
heightened political tension between them. In Subsection 6.1, we introduce the data on countries’ bilateral political
attitudes towards one another. In Subsection 6.2, we examine the relationship between these bilateral political attitudes
measures and our economic exposure measures.
6.1 Measuring Bilateral Political Attitudes
Building on a large literature in political science, we measure bilateral political attitudes between countries using
two di�erent data sources. First, we use voting behavior in the United Nations General Assembly (UNGA) to reveal
the bilateral similarity of countries’ foreign policies. Second we use measures of strategic rivalries, as classi�ed by
Thompson (2001) and Colaresi et al. (2010), based on contemporary perceptions by political decision makers of whether
41
countries regard one another as competitors, a source of actual or latent threats, or enemies.
6.1.1 United Nations General Assembly Votes
The ultimate source for our UN voting data is Voeten (2013), which reports non-unanimous plenary votes in the United
Nations General Assembly (UNGA) from 1962-2012, and includes on average around 128 votes each year. We use the
version of these data from the Chance-Corrected Measures of Foreign Policy Similarity (FPSIM) database, as reported
in Häge (2017).13 Countries are recorded as either voting “no” (coded 1), “abstain” (coded 2) or “yes” (coded 3).
We use several measures of bilateral voting similarity constructed from these data, as discussed in further detail in
Section G.2.1 of the online appendix. Our �rst and simplest measure is theS-score of Signorino and Ritter (1999), which
equals one minus the sum of the squared actual deviation between a pair of countries’ votes scaled by the sum of the
squared maximum possible deviations between their votes. By construction, this S-score measure of bilateral voting
similarity is bounded between minus one (maximum possible disagreement) and one (maximum possible agreement).
A limitation of this S-score measure is that is does not control for properties of the empirical distribution function
of country votes. In particular, country votes may align by chance, such that the frequency with which any two
countries agree on a “yes” depends on the frequency with which each country individually votes “yes.” Therefore, we
also consider two alternative measures of bilateral voting similarity that control in di�erent ways for properties of
the empirical distribution of votes. First, the π-score of Scott (1955) adjusts the observed variability of the countries’
voting similarity using the variability of each country’s own votes around the average vote for the two countries taken
together. Second, the κ-score of Cohen (1960) adjusts this observed variability of the countries’ voting similarity with
the variability of each country’s own votes around its own average vote.
Both the π-score and κ-score have an attractive statistical interpretation, as discussed further in Krippendorf
(1970), Fay (2005) and Häge (2011). In the case of binary (0,1) voting outcomes, these indices reduce to the form of
1 − (Do/De), where Do is the observed frequency of agreement and De is the expected frequency of agreement.
The key di�erence between the two indices is in their assumptions about the expected frequency of agreement. The
π-score estimates the expected frequency of agreement using the average of the two countries’ marginal distributions
of votes. In contrast, the κ-score estimates the expected frequency of agreement using each country’s own individual
marginal distribution of votes. All three of these measures of foreign policy similarity are necessarily symmetric,
whereas our economic measures of exposure to productivity shocks are potentially asymmetric, because country n’s
exposure to country i is not necessarily the same as i’s exposure to n.
Finally, as Bailey et al. (2017) point out, measures based on dyadic similarity of vote choices—such as the S, π,
and κ scores—do not account for the heterogeneity in resolutions being voted on. As a result, these measures could
incorrectly attribute changes in agenda as changes in state preferences. To resolve this issue, Bailey et al. (2017) apply
spatial voting models from the roll call literature to estimate each country’s political preferences embedded in its
UN votes. The outcome of this statistical procedure is a time-varying, one-dimensional measure called "ideal points",
which re�ects each country’s preference.14 Bailey et al. (2017) show that ideal points consistently capture the position
of states vis-à-vis a US-led liberal order. We derive a measure of bilateral distance by taking the absolute di�erence13See https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/ALVXLM14Speci�cally, Bailey et al. (2017)’s methodology identi�es preference change over time by exploiting duplicate resolutions that are voted repeat-
edly in consecutive sessions. This methodology also weights resolutions based on how much they re�ect the main policy preference dimension inorder to ensure that ideal points are not heavily in�uenced by resolutions that re�ect idiosyncratic factors.
42
between the ideal points of countries i and j in year t.
6.1.2 Strategic Rivalry
Our second set of measures of countries’ bilateral attitudes are indicator variables that pick up whether country i is a
strategic rival of j in year t, as classi�ed by Thompson (2001) and Colaresi et al. (2010). These rivalry measures capture
the risk of con�ict with a country of signi�cant relative size and military strength, based on contemporary perceptions
by political decision makers, gathered from historical sources on foreign policy and diplomacy. Speci�cally, rivalries
are identi�ed by whether two countries regard each other as competitors, a source of actual or latent threats that pose
some possibility of becoming militarized, or enemies.15
Prior research has shown that rivalry occurs much more frequently than actual wars (Colaresi et al. 2010, Aghion
et al. 2018). In our sample from 1970-2012, a total of 42 countries have had at least one strategic rival; 74 country-pairs
have been strategic rivals at some point; and the total number of country-pair-years that exhibit strategic rivalry is
2,452. China, for instance, is classi�ed as being in strategic rivalry with the U.S. (1970–1972 and 1996–present), India
(the entire sample period), Japan (1996–present), the former Soviet Union (1970–1989), and Vietnam (1973–1991).
6.2 Bilateral Political Attitudes and Economic Exposure
We now examine the relationship between these measures of bilateral political attitudes and our friends-and-enemies
su�cient statistics. We estimate the following regression speci�cation for importer n, exporter i, and time t:
Anit = βUnit + ηni + dt + εnit, (59)
where Anit is one of our measures of bilateral foreign policy similarity; Unit is our friends-and-enemies welfare
exposure measure; ηni is an exporter-importer �xed e�ect that controls for time-invariant unobserved heterogeneity
that is speci�c to an exporter-importer pair and a�ects both political attitudes and economic exposure; dt are year
dummies; in some speci�cations, we replace the year �xed e�ects with exporter-year �xed e�ects and importer-year
�xed e�ects; the exporter-year �xed e�ects control for the observation for which we shock productivity and capture
unobserved shocks that a�ect an exporter’s welfare exposure and political attitudes across all importers in a given
year; the importer-year �xed e�ects control for unobserved shocks that a�ect an importer’s welfare exposure and
political attitudes across all exporters in a given year; εnit is a stochastic error; and we cluster the standard errors in
all speci�cations by exporter-importer pair to allow for serial correlation in this error term over time.
Our inclusion of exporter-importer and year �xed e�ects implies that the regression speci�cation (59) has a
di�erence-in-di�erence interpretation, where the �rst di�erence is over time and the second di�erence is between
exporter-importer pairs. The key coe�cient of interest β is identi�ed from di�erential changes within exporter-
importer pairs over time: we examine whether as an exporter-importer pair becomes more economically friendly in
terms of the welfare e�ects of productivity growth, it also becomes more politically friendly in terms of foreign policy
similarity. A concern about estimating equation (59) using OLS is the potential for reverse causality: unobserved
changes over time in countries’ bilateral attitudes towards one another in the error term (εnit) could lead to changes
in bilateral trade between them. These changes in bilateral trade could in turn a�ect bilateral welfare exposure (Unit),15Colaresi et al. (2010) further re�ne the data to distinguish between three types of rivalries: spatial, where rivals contest the exclusive control of
a territory; positional, where rivals contest relative shares of in�uence over activities and prestige within a system or subsystem; and ideological,where rivals contest the relative virtues of di�erent belief systems relating to political, economic or religious activities.
43
thereby inducing a correlation between welfare exposure and the error term. To address this concern, we use a spec-
i�cation that follows Frankel and Romer (1999) in using predicted trade �ows abstracting from this variation as an
instrument. In particular, we estimate the following gravity equation for bilateral trade (xnit) between countries for
each year separately:
xnit = χntξitdistφtni$nit, (60)
where χnt is an importer-year �xed e�ect; ξit is an exporter-year �xed e�ect; φt is the time-varying coe�cient
on distance; and $nit is a stochastic error. We estimate this gravity equation using the Poisson Pseudo Maximum
Likelihood estimator of Santos Silva and Tenreyro (2006), which yields theory-consistent estimates of the �xed e�ects,
as shown in Fally (2015). To abstract from changes over time in countries’ bilateral attitudes for one another, we
use the �tted values from this regression (xnit = χntξitdistφtni) to construct predicted expenditure shares (snit =
xnit/∑Nm=1 xmit), thereby removing the bilateral error term $nit. In our class of models, these expenditure shares
(snit) determine income shares (tint), cross-substitution (mint), and hence welfare exposure (Unit). Therefore, we use
the predicted expenditure shares (snit) to instrument welfare exposure (Unit) in equation (59). Even after conditioning
on importer-year and exporter-year �xed e�ects, there is bilateral variation over time in these predicted expenditure
shares (snit), because of the time-varying coe�cient on distance (φt).
In Table 1, we report results of estimating equation (59) using both sets of attitudes measures (UN voting and
strategic rivalries) and our welfare exposure measure. We estimate this relationship using two-stage least squares,
instrumenting welfare exposure using our predicted trade shares. Columns (1)-(2) use the S-score; Columns (3)-(4)
use the π-score; Columns (5)-(6) use the κ-score; Columns (7)-(8) use the distance in ideal points; and Columns (9)-
(10) use strategic rivalries (all types). In each of these pairs of speci�cations, the �rst column ((1), (3), (5), (7) and (9))
includes only the exporter-importer and year �xed e�ects; the second column ((2), (4), (6), (8) and (10)) augments this
speci�cation with exporter-year and importer-year �xed e�ects. Panel A uses welfare exposure from the single-sector
model from Section 3 above; Panel B uses welfare exposure from the multi-sector model from Section 4.4 above; and
Panel C uses welfare exposure from the input-output model from Section 4.5 above.
Across Columns (1)-(6), we �nd positive and statistically signi�cant coe�cients in all speci�cations, implying
that as countries become greater economic friends in terms of the welfare e�ects of their productivity growth, they
also become greater political friends in terms of their voting similarity in the UNGA. Consistent with these results,
in Columns (7)-(10), we �nd negative and statistically signi�cant coe�cients in all speci�cations, implying that as
countries become greater economic friends in terms of the welfare e�ects of their productivity growth, they again
become greater political friends in terms of having smaller bilateral distances from the US-led liberal order and a
lower propensity to be strategic rivals. Beneath the coe�cient and standard error for each speci�cation, we report the
�rst-stage F-statistic. These �rst-stage F-statistics take the same value across Columns (1), (3) and (5) and Columns (2),
(4) and (6), because the �rst-stage regression speci�cation (welfare exposure on the instrument) and sample size is the
same across the di�erent political attitudes measures used in the second-stage regression. Although these �rst-stage
F-statistics naturally fall in the speci�cations including importer-year and exporter-year �xed e�ects, they remain
well above the conventional threshold of 10.16
16In Section G.2.2 of the online appendix, we show that we �nd a similar pattern of results for strategic rivalries when we consider the di�erenttypes of strategic rivalry separately (spatial, positional and ideological). Therefore, this relationship between the similarity of economic and politicalinterests holds regardless of which of these di�erent dimensions of strategic rivalries we consider.
44
T abl
e1:
Bila
tera
lPol
itica
lAtti
tude
s(Vo
ting
Sim
ilarit
yan
dSt
rate
gicR
ival
ry)a
ndW
elfa
reEx
posu
reT a
ble
1:Po
sitiv
ew
elfa
reex
posu
repr
edic
tsvo
ting
simila
rity
inUN
GAan
dlo
wer
likel
ihoo
dof
stra
tegi
criv
alry
(2SL
S)
Polit
ical
Out
com
eVo
ting
Sim
ilarit
y-S
Votin
gSi
mila
rity-κ
Votin
gSi
mila
rity-π
Dist
ance
inid
ealp
oint
sSt
rate
gicr
ival
ry(a
nyty
pe)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Pane
l A:W
elfa
reex
posu
rein
singl
e-se
ctor
mod
el
USin
gle−secto
r9.7
36∗∗
∗11
.78∗∗
∗23
.09∗∗
∗21
.54∗∗
∗26
.09∗∗
∗24
.79∗∗
∗-3
7.26∗
∗∗-3
2.97∗
∗∗-4
.741∗
∗∗-5
.379∗
∗∗
(2.73
8)(2
.446)
(4.64
4)(4
.434)
(5.40
3)(4
.967)
(10.8
2)(8
.535)
(1.78
2)(1
.960)
Firs
t-sta
geF
98.08
78.50
98.08
78.50
98.08
78.50
110.4
86.86
99.50
78.52
Pane
lB:W
elfa
reex
posu
rein
mul
ti-se
ctor
mod
el
UM
ulti−
secto
r9.6
35∗∗
∗11
.66∗∗
∗22
.85∗∗
∗21
.32∗∗
∗25
.82∗∗
∗24
.54∗∗
∗-3
6.87∗
∗∗-3
2.64∗
∗∗-4
.695∗
∗∗-5
.331∗
∗∗
(2.71
0)(2
.428)
(4.61
0)(4
.407)
(5.35
6)(4
.933)
(10.7
3)(8
.487)
(1.76
6)(1
.944)
Firs
t-sta
geF
98.61
78.76
98.61
78.76
98.61
78.76
110.4
86.67
99.36
78.20
Pane
lC:W
elfa
reex
posu
rein
inpu
t-out
putm
odel
UInput−
Outp
ut
20.41
∗∗∗
26.14
∗∗∗
48.42
∗∗∗
47.80
∗∗∗
54.69
∗∗∗
55.00
∗∗∗
-76.9
4∗∗∗
-72.1
9∗∗∗
-9.83
1∗∗∗
-11.8
3∗∗∗
(5.21
1)(4
.638)
(7.96
5)(8
.192)
(9.37
9)(9
.156)
(20.2
4)(1
6.74)
(3.49
5)(4
.054)
Firs
t-sta
geF
161.1
113.6
161.1
113.6
161.1
113.6
163.5
114.3
173.6
119.8
Spec
i�ca
tion:
2SLS
Exp×
Imp
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Year
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Exp×
Year
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
Imp×
Year
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No.
ofO
bs.
5858
8458
5884
5858
8458
5884
5858
8458
5884
5677
9056
7790
6109
5461
0954
No.
ofCl
uste
rs14
721
1472
114
721
1472
114
721
1472
114
479
1447
914
761
1476
1
Stan
dard
erro
rsin
pare
nthe
sesa
recl
uste
red
atco
untr
y-pa
irle
vel
∗p<
0.1
,∗∗p<
0.05
,∗∗∗p<
0.01
245
Taken together, these empirical results are consistent with the view that increased con�ict of economic interests
between countries leads to heightened political tension between them. As these measures of bilateral political attitudes
are measured entirely independently of any of these quantitative trade models, they also provide an external validation
that our friends-and-enemies measures of welfare exposure are systematically related to independent measures of
countries’ attitudes towards one another.
7 Conclusions
The closing decades of the twentieth century saw large-scale changes in the relative economic size of nations, with
China’s rapid economic growth transforming it into a major trading nation. A classic question in international eco-
nomics is the implications of such economic growth for the income and welfare of trade partners. A related question
in political economy is the extent to which these large-scale changes in relative economic size necessarily involve
political tension and realignments. We provide new theory and evidence on both of these questions by developing
bilateral “friends” and “enemies” measures of countries’ income and welfare exposure to foreign productivity shocks
that can be computed using only observed trade data. We show that these measures are exact for small productivity
shocks in the leading class of international trade models characterized by a constant trade elasticity. For large produc-
tivity shocks, we characterize the quality of the approximation in terms of the properties of the observed trade data,
and show that for the magnitude of the productivity shocks implied by the observed data, our exposure measures
are almost visibly indistinguishable from the predictions of the full non-linear solution of the model. Our approach
admits a large number of extensions and generalizations, including multiple sectors, input-output linkages and eco-
nomic geography (factor mobility). Using our methods, we derive bounds for the sensitivity of countries’ exposure to
foreign productivity shocks to departures from a constant trade elasticity.
We contribute to the recent revolution in international trade of the development of quantitative trade models. A
key advantage of these quantitative models is that they are rich enough to capture �rst-order features of the data,
such as a gravity equation for bilateral trade, and yet remain su�ciently tractable as to be amenable to counterfactual
analysis, with a small number of structural parameters. A key challenge is that these models are highly non-linear,
which can make it di�cult to understand the economic explanations for quantitative �ndings for particular countries
or industries. A key contribution of our bilateral friends-and-enemies measures is to allow researchers to connect
quantitative results to the key underlying economic mechanisms in the model: the cross-substitution e�ect, where an
increase in the competitiveness of a foreign country leads consumers in all markets to substitute away from all other
nations; a market-size e�ect, where an increase in income in foreign markets raises demand for all nations’ goods;
and a cost-of-living e�ect, where an increase in the competitiveness of a country’s goods reduces the cost of living in
all countries. As our linearization uses standard matrix inversion techniques, we �nd that it is around 70,000 faster
than solving the full non-linear solution of the model. Therefore, our methods are well suited to applications where
large numbers of counterfactuals are required, and facilitate comparisons of these counterfactuals across alternative
quantitative frameworks, such as our single-sector, multi-sector and input-output models.
We use our friends-and-enemies exposure measures to examine the global incidence of productivity growth in
each country on income and welfare for more than 140 countries over more than 40 years from 1970-2012 (around
one million bilateral comparative statics). We �nd a substantial and statistically signi�cant increase in both the mean
46
and dispersion of welfare exposure to foreign productivity growth over our sample period, consistent with increas-
ing globalization enhancing countries’ economic dependence on one another. We also observe large-scale changes in
bilateral networks of welfare exposure between nations. We �nd that the cross-substitution, market-size and cost-of-
living e�ects are all quantitatively important for the welfare impact of foreign productivity growth, and the general
equilibrium forces in this class of quantitative trade models are large relative to the partial equilibrium e�ects of pro-
ductivity growth. Whether we consider the single-sector, multi-sector or input-output models, we �nd that Chinese
productivity growth has reduced aggregate U.S. income, both relative to world GDP and other OECD countries. Nev-
ertheless, we �nd that Chinese productivity growth has raised aggregate U.S. welfare, highlighting the strength of the
cost-of-living e�ect in these quantitative trade models. Consistent with the idea that con�icting economic interests
can spawn political discord, we �nd that as countries become greater economic friends in terms of the welfare e�ects
of their productivity growth, they become greater political friends in terms of the similarity of their foreign policy
stances, as measured by United Nations voting patterns and strategic rivalries.
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